2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions

We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels--Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen--Stoll. We establish the local formula in many instances and show that in remaining cases it follows from standard global conjectures.

where w(A/K) ∈ {±1} is the global root number of A/K. The Birch and Swinnerton-Dyer conjecture asserts that the Mordell-Weil rank of A/K agrees with the order of vanishing at s = 1 of L ⋆ (A/K, s): ord s=1 L ⋆ (A/K, s)=rk(A/K). If w(A/K) = 1 (resp. −1), then L ⋆ (A/K, s) is an even (resp. odd) function around s = 1 and as such its order of vanishing there is even (resp. odd). Thus a consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: Essentially all progress towards the parity conjecture has proceeded via the p-parity conjecture. For a fixed prime p, denote by rk p (A/K) the p-infinity Selmer rank of A/K. Under the conjectural finiteness of the Shafarevich-Tate group (or indeed, under the weaker assumption that its p-primary part is finite), rk p (A/K) agrees with rk(A/K). The p-parity conjecture is the assertion that w(A/K) = (−1) rkp(A/K) .
Note that without knowing finiteness of the Shafarevich-Tate group, these conjectures are inequivalent for different primes p.
1.1. Known cases of the p-parity conjecture. For elliptic curves over Q, Dokchitser-Dokchitser [DD10] have shown that the p-parity conjecture holds for all primes p. Subsequently, Nekovář [Nek13] extended this result to all totally real number fields, excluding some elliptic curves with potential complex multiplication; these exceptional cases have recently been treated by Green-Maistret [GM21]. For a general number field K, Česnavičius [Čes16] has shown that the p-parity conjecture holds for elliptic curves over K possessing a p-isogeny, whilst work of Kramer-Tunnell [KT82] and Dokchitser-Dokchitser [DD11] proves that the 2-parity conjecture holds for an arbitrary elliptic curve E/K, not over K itself, but over any quadratic extension of K.
For higher dimensional abelian varieties much less is known. The most general result at present is due to Coates, Fukaya, Kato and Sujatha, who prove in [CFKS10] that for odd primes p, the p-parity conjecture holds for any abelian variety possessing a suitable p-power degree isogeny, subject to some further technical conditions. For p = 2 the main result is due to Dokchitser-Maistret [DM19], who prove the 2-parity conjecture for quite general semistable abelian surfaces.
1.2. Main result. Following on from the work of Kramer-Tunnell and Dokchitser-Dokchitser for elliptic curves, we consider the 2-parity conjecture for Jacobians of hyperelliptic curves over quadratic extensions of their field of definition. Our main result is the following: Theorem 1.1. Let K be a number field and L/K a quadratic extension. Let C/K be a hyperelliptic curve of genus g ≥ 2 and let J/K be the Jacobian of C. Suppose that J has semistable reduction at each prime p ∤ 2 of K which ramifies in L/K, and assume moreover that: • for each prime p | 2 of K which is inert in L/K, J has good reduction at p, • for each prime p | 2 of K which ramifies in L/K, J has good ordinary reduction at p and K p (J[2])/K p has odd degree. Then the 2-parity conjecture holds for J/L. Remark 1.2. Theorem 1.1 gives a large supply of hyperelliptic curves satisfying the 2-parity conjecture over every quadratic extension of their field of definition; see Lemma 16.5 for explicit conditions on a Weierstrass equation defining C that ensure the conditions of Theorem 1.1 at primes dividing 2 are satisfied. Remark 1.3. If the genus of C is 2 then one can weaken the assumption that J has good reduction at each inert prime dividing 2 to assume only that J has semistable reduction at such primes; see Proposition 9.1.
1.3. Reduction to a local question. The proof of Theorem 1.1 proceeds by reducing to a purely local question, as we now explain.
In the notation of Theorem 1.1, for each place v of K which is non-split in L, denote by v the unique place of L extending v. Since J is defined over K, the root number w(J/L) decomposes as a product of local terms indexed by places of K which are non-split in L/K: where w(J/L v ) ∈ {±1} is the local root number of J/L v . The strategy to prove Theorem 1.1 is to similarly decompose the parity of the 2-infinity Selmer rank of J over L into local terms, and compare these place by place. Specifically, results of [Mor19] combined with work of Poonen-Stoll [PS99] give a decomposition of the parity of rk 2 (J/L) into local terms as detailed below, generalising a theorem of Kramer [Kra81, Theorem 1] for elliptic curves. Before stating this decomposition we need to introduce some notation. Note that, as a quotient of J(K v )/2J(K v ), the cokernel of this map is a finite dimensional F 2 -vector space.
Define also the invariant ǫ(C/K v ) ∈ {0, 1} by setting Here, following [PS99, Section 8], we say that C/K v is deficient if C has no K v -rational divisor of degree g − 1.
The relevance of the invariant ǫ(C/K v ) comes from a result of Poonen and Stoll [PS99, Theorem 8] characterising the failure of the Shafarevich-Tate group of J/K to have square order (if finite) in terms of the ǫ(C/K v ). Denoting by C L /K the quadratic twist of C by L, we define ǫ(C L /K v ) similarly. We then have the following decomposition of the parity of rk 2 (J/L) into local terms: Theorem 1.6 (=Theorem 2.1). We have Ideally, one might hope that the local terms contributing to w(J/L) and (−1) rk 2 (J/L) simply agree place by place. However, this turns out not to be the case, and so the strategy hinges on identifying the discrepancy between these local terms as a quantity which vanishes globally. To this end we conjecture the following, generalising a formula of Kramer-Tunnell [KT82] for elliptic curves: Conjecture 1.7. Let K be a local field of characteristic different from 2. Let L/K be a quadratic extension, let C/K be a hyperelliptic curve, and denote by J/K the Jacobian of C. Then we have w(J/L) = (∆ C , L/K)(−1) ǫ(C/K)+ǫ(C L /K)+dim J(K)/N L/K J(L) .
Here the quantity ∆ C is the discriminant of f (x) for any Weierstrass equation y 2 = f (x) defining C, and (∆ C , L/K) ∈ {±1} is the Hilbert/Artin symbol of ∆ C with respect to the extension L/K. 1 Returning now to the case where L/K is a quadratic extension of number fields and C is a hyperelliptic curve defined over K, by the product formula for Hilbert symbols we have v place of K v non-split in L (∆ C , L v /K v ) = 1.
In particular, we see from (1.4) and Theorem 1.6 that Conjecture 1.7 implies the 2-parity conjecture for J/L. We will prove Conjecture 1.7 under the assumptions on the reduction of C appearing in the statement of Theorem 1.1, hence proving that result. Specifically, our second main result is the following: Theorem 1.8. Conjecture 1.7 holds in the following cases: • K = R, • K has odd residue characteristic, and either L/K is unramified or J/K has semistable reduction, • K is a finite extension of Q 2 , L/K is unramified, and either J/K has good reduction or g = 2 and J/K has semistable reduction, • K is a finite extension of Q 2 , J/K has good ordinary reduction, and K(J[2])/K has odd degree.
Remark 1.9. More generally, Conjecture 1.7 holds if there is an odd degree Galois extension F/K over which C satisfies the conditions of Theorem 1.8 with L/K replaced by F L/F ; see Section 4.
As further evidence for Conjecture 1.7, we show that the cases above (and in fact substantially fewer) are sufficient to deduce Conjecture 1.7 from the 2-parity conjecture via a global-to-local argument, at least for curves arising via base-change from a number field. Theorem 1.10 (=Theorem 8.1). Let K be a number field, C/K a hyperelliptic curve, J/K its Jacobian, and v 0 a place of K. If the 2-parity conjecture holds for J over every quadratic extension of K, then Conjecture 1.7 holds for J/K v 0 and every quadratic extension L/K v 0 .
Remark 1.11. We remark that Conjecture 1.7 makes sense (and, surprisingly, is not entirely vacuous) in genus 0. Indeed, for a quadratic extension L/K of local fields of characteristic different from 2, consider a hyperelliptic curve C : is a squarefree polynomial of degree 1 or 2. The Jacobian of C is trivial, so the root number and cokernel of the local norm map are trivial also. Further, C/K (resp. C L /K) is deficient if and only if it has no K-point. It is then easy to check that (∆ C , L/K) = (−1) ǫ(C/K)+ǫ(C L /K) for any quadratic extension L/K.

1.4.
Comparison with work of Kramer-Tunnell. Conjecture 1.7 has its origins in work of Kramer-Tunnell. Specifically, for a local field K, a separable quadratic extension L/K, and an elliptic curve E/K, Kramer-Tunnell [KT82] conjectured the formula (1.12) w(E/K)w(E L /K) = (−∆ E , L/K)(−1) dim E(K)/N L/K E(L) , and proved it in many cases, including in every instance when K has odd residue characteristic. This conjecture is now known in all cases thanks to subsequent work of Dokchitser-Dokchitser [DD11] and Česnavičius-Imai [ČI16]. By [Čes16,Proposition 3.11] we have w(E/L) = w(E/K)w(E L /K)(−1, L/K), whilst ǫ(E/K) = 0 for every local field K and elliptic curve E/K. Thus Conjecture 1.7 specialises to (1.12) when C/K is an elliptic curve. The presence of the new terms ǫ(C/K) and ǫ(C L /K) in the purely local Conjecture 1.7, which are 'forced' by global considerations concerning the possible failure of the Shafarevich-Tate group of a principally polarised abelian variety to have square order (see Section 2), is a key new feature of this work. These terms also place constraints on possible proofs of Conjecture 1.7. Indeed, ǫ(C/K) is not a function purely of the Jacobian of C (as in Remark 1.11, ǫ(C/K) can be non-trivial even for curves of genus 0!). A lot of the technical difficulty in this work is involved in relating invariants defined in terms of the Jacobian of C, such as the cokernel of the local norm map, to the invariants ǫ(C/K), ǫ(C L /K) and (∆ C , L/K), which have no obvious meaning for general abelian varieties.
As above, the Kramer-Tunnell formula (1.12) is known to hold for local fields of characteristic 2 and separable quadratic extensions L/K. It is thus tempting to extend the scope of Conjecture 1.7 to include such extensions (especially in light of the work of Česnavičius-Imai [ČI16] who reduce (1.12) over local fields of characteristic 2 to the corresponding conjecture for finite extensions of Q 2 ). However, since we prove no instances of Conjecture 1.7 over local fields of characteristic 2 in this work, we have elected not to do this.
1.5. Overview of the paper. In Section 2 we explain how to deduce Theorem 1.6 by combining results of [Mor19] with work of Poonen-Stoll [PS99].
In Section 3 we recall and prove some basic properties of the local norm map for general abelian varieties. Of particular use later is Lemma 3.4 which, for nonarchimedean local fields of odd residue characteristic, expresses the order of the cokernel of the local norm map in terms of Tamagawa numbers, generalising a result of Kramer-Tunnell [KT82, Corollary 7.6] for elliptic curves.
In Section 4 we prove some compatibility results concerning the behaviour of Conjecture 1.7 under quadratic twist, and under odd-degree Galois extension of the base field.
Across Sections 5 and 6 we collect and prove some basic results concerning, respectively, 2torsion in Jacobians of hyperelliptic curves, and criteria for determining when a hyperelliptic curve over a local field K is deficient. Whilst much of this material is standard, Proposition 6.7, which characterises deficiency for a particular class of hyperelliptic curves (essentially those with a Krational theta characteristic) may be of independent interest.
In Section 7 we combine the results of Sections 5 and 6 to deduce some simple cases of Conjecture 1.7. Namely, we establish Conjecture 1.7 when K is nonarchimedean, and when K has odd residue characteristic and J/K has good reduction. Then in Section 8 we show that these special cases are already enough to deduce Theorem 1.10.
With the exception of the short Sections 16 and 17 (which, respectively, consider Conjecture 1.7 for finite extensions of Q 2 , and tie together results from previous sections to prove Theorems 1.1 and 1.8), the remainder of the paper splits into 2 parts. Firstly, in Sections 9 and 10 we consider Conjecture 1.7 when the extension L/K is unramified, proving it completely in this case when K has odd residue characteristic. We do this by analysing the minimal proper regular model of C. The key fact making Conjecture 1.7 accessible here is that the formation of the minimal regular model commutes with unramified base change; this enables a comparison between invariants of C and those of its unramified quadratic twist. The central technical result of these sections is Theorem 10.2, which we formulate for general curves, and which shows that the quantity viewed as an element of Q × /Q ×2 , behaves well under quite general twisting. Here k is the residue field of K and Φ is the Néron component group of the Jacobian of C. We would also like to advertise Proposition 10.8, which is a by-product of the proof of Theorem 10.2, and which gives a relatively simple way of computing the Tamagawa number of the Jacobian of an arbitrary curve, modulo rational squares, as a function of its minimal regular model. This result plays a prominent role in simplifying computations in Section 14. Finally, across Sections 11 to 15, we prove Conjecture 1.7 when C/K has semistable reduction and when L/K is a ramified quadratic extension of local fields with odd residue characteristic. Roughly speaking, once again our strategy is to encode each of the invariants appearing in Conjecture 1.7 in terms of the minimal proper regular models of both C and C L . However, since now L/K is ramified, the minimal regular model of C L can be significantly different to that of C, making it hard to relate the relevant invariants. We overcome this by fixing a Weierstrass equation y 2 = f (x) for C and drawing on the explicit description of the minimal regular models of C and C L in terms of clusters (certain combinatorial objects encoding the distances between the roots of f (x)) afforded by the works [DDMM18] and [FN20]. This essentially reduces Conjecture 1.7 to a purely combinatorial question about clusters, though one that still seems far from straightforward. We split the resulting analysis into two parts. First, in Proposition 13.20 we give an explicit description in terms of clusters of the group B C/K introduced by Betts-Dokchitser in [BD19]; this group packages together information about the Tamagawa number of the Jacobian of C over both K and L, but seems simpler to describe than each of these quantities. Then in Section 14 we study the minimal regular model of C L , describing in terms of clusters the Tamagawa number of the Jacobian of C L modulo rational squares; see Corollary 14.30. Finally, in Section 15 we combine these results to establish the sought case of Conjecture 1.7.
Notation and conventions. For a field K we denote byK a (fixed once and for all) algebraic closure of K, and denote by K s ⊆K the separable closure of K. We denote by G K = Gal(K s /K) the absolute Galois group of K.
1.5.1. Hyperelliptic curves. By a hyperelliptic curve C over a field K we mean a smooth, proper, geometrically connected curve of genus g ≥ 2, defined over K, and admitting a finite separable morphism C → P 1 K of degree 2. When K has characteristic different from 2, one can always find a separable polynomial f (x) ∈ K[x] of degree 2g + 1 or 2g + 2 such that C is isomorphic to the curve given by gluing the affine schemes via the relations x = 1/u and y = x g+1 v. By an abuse of notation we say that C is given by the Weierstrass equation y 2 = f (x), and refer to elements of U 2 (K) U 1 (K) as the points at infinity.
There are 2 such points if deg(f ) is even, and 1 if deg(f ) is odd. We denote by ι the hyperelliptic involution of C. For C : y 2 = f (x) this is the automorphism (x, y) → (x, −y). When char(K) = 2, we define the discriminant ∆ C ∈ K × of a hyperelliptic curve given by a Weierstrass equation C : y 2 = f (x) by the formula given in [Liu96, Seciton 2]. One sees from that work that, up to squares in K × , this both agrees with the polynomial discriminant of f (x) and is independent of the choice of Weierstrass equation for C/K. In particular, we will often consider ∆ C ∈ K × /K ×2 without reference to a Weierstrass equation for C. Further, if we write , hence agree modulo squares in K. In particular, the class 1.5.2. Quadratic twists. Let K be a field of characteristic different from 2, and L/K a quadratic extension. For a hyperelliptic curve C/K we denote by C L /K the quadratic twist of C by L/K. This is the twist of C/K corresponding to the 1-cocycle Suppose that C/K is given by a Weierstrass equation y 2 = f (x), and that L = K( √ d) for some d ∈ K × . Then C L /K is given by the Weierstrass equation y 2 = df (x). In particular, it follows from the discussion on hyperelliptic discriminants above that, as elements of K × /K ×2 , we have For an abelian variety A/K we similarly denote by A L /K the quadratic twist of A by L/K, which corresponds to the 1-cocycle Denote by χ : G K → {±1} the quadratic character corresponding to L/K. Then there is a K sisomorphism ψ : A ∼ −→ A L such that, for all σ ∈ G K , the composition ψ −1 • σ ψ is multiplication by χ(σ) on A, where σ ψ denotes the unique isomorphism A → A L acting as σ • ψ • σ −1 on K s -points. In particular, ψ restricts to an isomorphism of Since the hyperelliptic involution on C induces multiplication by −1 on its Jacobian J/K, the Jacobian of C L /K coincides with J L /K. 1.5.3. Galois cohomology. For a profinite group G, a discrete G-module M , and integer i ≥ 0, we denote by H i (G, M ) the i-th cohomology group of G with coefficients in M , as defined in e.g. [Gru67]. We denote by M G the subgroup of elements of M fixed by G. For g ∈ G we denote by M g the subgroup of elements fixed by g.
When G = G K for a field K we will often write H i (K, M ) in place of H i (G K , M ). Similarly, for a Galois extension L/K and a discrete Gal(L/K)-module M , we often write H i (L/K, M ) in place of H i (Gal(L/K), M ).
1.5.4. Notation for number fields and local fields. For a number field K we denote by O K the ring of integers of K. For a place v of K, K v will denote the corresponding completion.
By a local field K we mean a locally compact valued field. Thus K is isomorphic (as a valued field) to one of R, C, or a finite extension of either Q p or F p ((t)) for a prime p. For a nonarchimedean local field K we take the following notation: ring of integers of K, k residue field of K, π a choice of uniformiser of K, v :K × → Q valuation onK normalised with respect to K, so that v(π) = 1, m maximal ideal of the ring of integers ofK, K nr maximal unramified extension of K, (a, L/K) Artin symbol of a ∈ K × in a Galois extension L/K. We will usually take L/K quadratic, in which case we regard this symbol as being valued in {±1}.
1.5.5. Notation for curves and abelian varieties. For a smooth, proper, geometrically connected curve X over a local field K, we define ǫ(X/K) ∈ {0, 1} to be equal to 1 if X is deficient over K, and equal to 0 else. Thus ǫ(X/K) = 1 if and only if X has a K-rational divisor of degree g − 1, where g is the genus of X. Throughout the paper, for a field K, C/K will almost always denote a hyperelliptic curve over K, g will denote the genus of C, and J/K will denote the Jacobian of C.
For an abelian variety A over a field K (usually the Jacobian of a hyperelliptic curve C), we take the following notation.
For K a number field: the Shafarevich-Tate group of A/K, X nd (A/K) the quotient of X(A/K) by its maximal divisible subgroup, w(A/K) the global root number of A/K.
For K a nonarchimedean local field: Φ the component group of the special fibre of the Néron model of A/K; we often refer to this as the Néron component group of A. c(A/K) the Tamagawa number of A/K. By definition this is the order of the group Φ(k) of k-rational points of Φ, w(A/K) the local root number of A/K, N L/K for L/K separable quadratic, denotes the norm map A(L) → A(K) sending P ∈ A(L) to N L/K := σ∈Gal(L/K) σ(P ).

2-Selmer groups in quadratic extensions
In this section we combine results of [Mor19] and [PS99] to deduce Theorem 1.6. Let L/K be a quadratic extension of number fields, let C/K be a hyperelliptic curve, and let J/K denote the Jacobian of C. Further, denote by rk 2 (J/K) the 2-infinity Selmer rank of J/K, and recall from Notation 1.5 the definitions of the local norm map and the invariant ǫ(C/K v ) for a place v of K. Let C L /K (resp. J L /K) denote the quadratic twist of C (resp. J) by L/K. Theorem 2.1 (=Theorem 1.6). We have Proof. By [Mor19, Theorem 10.12] we have where here Sel 2 (J/K) denotes the 2-Selmer group of J/K (and similarly for J L /K). Consequently (cf. [Mor19, proof of Theorem 10.20]) we have where X nd (J/K) denotes the quotient of the Shafarevich-Tate group of J/K by its maximal divisible subgroup. It follows from [PS99, Theorem 11] that where both congruences are modulo 2. For the second equality we are using that the Jacobian of C L coincides with the quadratic twist J L . Since C and C L are isomorphic over K v for each place v that splits in L/K, the result follows.
Remark 2.2. One of the key reasons for working with Jacobians of hyperelliptic curves in this paper is that the quadratic twist J L is again the Jacobian of an explicit curve: the quadratic twist C L . This allows us to give an explicit description of the parity of both dim X nd (J/K)[2] and dim X nd (J L /K)[2] in terms of deficiency.

Basic properties of the local norm map
In this section we prove some basic properties of the cokernel of the local norm map. Take K to be a local field of characteristic different from 2, and let L/K be a quadratic extension. We work with arbitrary principally polarised abelian varieties since everything goes through in this setting. Thus for now we fix a principally polarised abelian variety A/K, and denote by A L the quadratic twist of A by L/K. Denote by N L/K : A(L) → A(K) the local norm map, sending P ∈ A(L) to σ∈Gal(L/K) σ(P ). Further, denote by χ : G K → {±1} the quadratic character corresponding to L/K.
as desired.
From the definition of the quadratic twist A L we have an L-isomorphism ψ : A ∼ −→ A L such that, for all σ ∈ G K , the composition ψ −1 • σ ψ is multiplication by χ(σ) on A. The map ψ −1 identifies A L (L) with A(L), and identifies A L (K) with ker N L/K : A(L) → A(K) . The local norm map A L (L) → A L (K) then identifies with the map sending P ∈ A(L) to P − σ(P ). To avoid confusion, we denote this map by N L L/K .
Proof. That A(K)/N L/K A(L) is a finite dimensional F 2 -vector space follows from the fact that 2A(K) ⊆ N L/K A(L) along with the well-known finiteness of A(K)/2A(K). Next, consider the map . This is readily checked to be a (well defined) isomorphism. The result now follows from Lemma 3.1. Now let Res L/K A denote the Weil restriction of scalars of A from L to K. This is an abelian variety over K of dimension 2dimA which represents the functor T → A(T × K L). As explained in [Mil72, Section 2] (see also [MRS07,Proposition 4.1]), denoting by γ the involution of A × A swapping the factors, Res L/K A can be described as the twist of A × A corresponding to the 1- This identifies (Res L/K A)(K) with the L-points of A diagonally embedded in A(K)×A(K), realising the functor of points description for T = K. As above, both Res L/K A and A × A L are twists of A × A, and one checks that the endomorphism of A × A given by (P, Q) → (P + Q, P − Q) descends to an isogeny φ : We exploit the isogeny φ to prove the final lemma of this section, which expresses the cokernel of the local norm map in terms of Tamagawa numbers. The special case of this for elliptic curves is due to Kramer and Tunnell [KT82, Corollary 7.6], although the proof is different. Recall from Section 1.5.5 that c(A/K) denotes the Tamagawa number of A/K. Proof. To ease notation write X = Res L/K A and Y = A × A L . With φ as above, since K has odd residue characteristic it follows from a formula of Schaefer [Sch96, Lemma 3.8] that the first map induced by inclusion into the second factor, and the second map being the projection onto the first factor. Since K has odd residue characteristic we have (cf. [Sch96, Proposition 3.9] for example) ). The result now follows by combining this last observation with (3.5) and (3.6).

Compatibility results
In this section we prove several compatibility results for Conjecture 1.7. These provide some evidence in favour of the conjecture and will also be used to make some reductions as part of the proof of Theorem 1.8.
In what follows, K denotes a local field of characteristic different from 2. Let L/K be a quadratic extension and let C/K be a hyperelliptic curve.

Odd degree Galois extensions.
Lemma 4.1. Every individual term in Conjecture 1.7 is unchanged under odd degree Galois extension of the base field. In particular, if F/K is an odd degree Galois extension, then Conjecture 1.7 holds for C/K and the extension L/K if and only if it holds for C/F and the extension LF/F . Proof. That the term (∆ C , L/K) is invariant under odd degree extensions (not necessarily Galois) is standard. Similarly, it's not hard to show that the terms involving deficiency of C and its twist are also individually invariant under arbitrary odd degree extensions (cf. Lemma 6.4 for a more general result which implies this). The statement for each of the root numbers is also standard; see for example [  Proof. Since the root number and terms involving deficiency appear symmetrically between J and J L in Conjecture 1.7, it suffices to show that For the first equality one checks readily that ∆ C and ∆ C L lie in the same class in K × /K ×2 (cf. Section 1.5.2). The second statement follows from Lemma 3.2.
4.3. Second compatibility with quadratic twist. The second compatibility result involving quadratic twist is more subtle. That such a compatibility result should exist for elliptic curves was discussed in the original paper of Kramer and Tunnell [KT82, remark following Proposition 3.3] and the result was later proven (again for elliptic curves) by Klagsbrun, Mazur and Rubin [KMR13,Lemma 5.6]. The key step in that proof is to establish the following congruence. In order to state it, fix distinct quadratic extensions L 1 /K and L 2 /K, and denote by L 3 /K the third quadratic subextension of L 1 L 2 /K. Lemma 4.3. Let A/K be a principally polarised abelian variety. Then we have Proof. The case where A/K is an elliptic curve is [KMR13, Lemma 5.6] and the argument is essentially the same. Let L 0 = K and for each i = 1, 2, 3 identify A L i [2] with A[2] as G K -modules in the usual way. For each i let X i denote the image of A L i (K)/2A L i (K) under the map where δ L i is the connecting homomorphism associated to the multiplication-by-2 Kummer sequence for A L i . By [MR07, Proposition 5.2], for i = 1, 2, 3 we have

Similarly, we have
In the elliptic curve case treated in [KMR13] it is shown that each X i is a maximal isotropic subspace with respect to a certain quadratic form on H 1 (K, A[2]). The result is then deduced from [KMR13, Corollary 2.5] which is a general result concerning the parity of the dimension of intersections of maximal isotropic subspaces. For general principally polarised abelian varieties, the fact that each X i is a maximal isotropic subspace for the natural generalisation of this quadratic form is detailed in [Mor19, Section 10.1]. The one difference from the case of elliptic curves is that now the quadratic form (in general) takes values in Z/4Z, rather than just Z/2Z as is assumed in [KMR13, Corollary 2.5]. However, one readily verifies that this assumption is not used in the proof of [KMR13, Corollary 2.5].
We now return to the case where C/K is a hyperelliptic curve and J/K is its Jacobian.
Corollary 4.4. Conjecture 1.7 for J/K and the extensions L 1 /K and L 2 /K implies Conjecture 1.7 for J L 1 /K and the extension L 3 /K. where g is the genus of C (the cited result is only stated for elliptic curves, but the proof generalises verbatim to give the claimed formula). From (4.5) it follows that Further, by standard properties of Hilbert symbols and the fact that the discriminants of C and any quadratic twist of C differ by squares, we have Since we also have the result follows from Lemma 4.3.
Remark 4.6. For a local field K and hyperelliptic curve C/K, it follows from Lemma 4.2 and Corollary 4.4 that, in order to prove Conjecture 1.7 for C/K and all quadratic extensions of K, it suffices to prove the same result but with C replaced by an arbitrary quadratic twist.

Two torsion in the Jacobian of a hyperelliptic curve
For this section let K be a field of characteristic different from 2. Let C/K : y 2 = f (x) be a hyperelliptic curve of genus g and let J/K be its Jacobian. Denote by W the G K -set of ramification points of the x-coordinate morphism C → P 1 . Thus W consists of the points (r, 0) for r a root of f (x), along with the unique point at infinity on C if deg(f ) is odd. As G K -modules we then have Here F W 2 denotes the permutation representation over F 2 on the elements of W, Σ : F W 2 → F 2 denotes the sum-of-coefficients map, and D = w∈W w. See [PS97, Section 6] for more details. Noting that g ≥ 2, hence |W| > 4, we see from the above description that K(J[2])/K is the splitting field of f (x).
We now compute the dimension of the rational 2-torsion J(K) [2]. The case where K(J[2])/K is cyclic is treated in [Cor01, Theorem 1.4] (but note the erratum [Cor05]) whilst the case where f (x) has an odd degree factor over K is [PS97, Lemma 12.9]. We will require a slightly more general statement.
In what follows we write is monic. To clean up the statement we also make the following convention.
Convention 5.2. In what follows, if deg(f ) is odd, the rational point at infinity on C is to be interpreted as an odd degree irreducible factor of f (x) over K. Otherwise, if each irreducible factor of f (x) has even degree, let F/K be the splitting field of f (x) and let m be the number of quadratic subextensions of F/K over which f 0 (x) factors as a product of 2 distinct conjugate polynomials. Then dim J(K)[2] = n − 1 g even, n − 1 + ord 2 (1 + m) g odd.
Proof. Denote by G the Galois group of F/K and let M be the G-module M = ker F W 2 Σ −→ F 2 . Then by (5.1) we have an exact sequence has an odd degree factor over K, n else.
Consequently, we must show that ker H 1 (G, F 2 D) → H 1 (G, M ) has dimension 0 or ord 2 (1 + m) according, respectively, to whether g is even or odd (note that if f (x) has an odd degree factor over K then m = 0). Now H 1 (G, F 2 D) = Hom(G, F 2 D), and the non-trivial homomorphisms from G into F 2 D correspond to the quadratic subextensions of F/K. Let φ be such a homomorphism, corresponding to a quadratic subextension E/K. Then φ maps to 0 in H 1 (G, M ) if and only if there is η ∈ M with σ(η) + η = φ(σ)D for each σ ∈ G. Now an element η ∈ F W 2 satisfying this equation corresponds to a factor of f 0 (x) over E, h(x) say, for which f 0 (x) = h(x) · τ h(x), where τ denotes the generator of Gal(E/K). Since 1 2 |W| = g + 1, such an η is in the sum-zero part of F W 2 if and only if g is odd. We conclude from this that the number of non-identity elements in ker H 1 (G, F 2 D) → H 1 (G, M ) is equal to 0 if g is even, and m if g is odd. This gives the result.
Now let ∆ f be the discriminant of f (x). It is a square in K if and only if the Galois group of f (x) is a subgroup of the alternating group A n where n = deg f . As a corollary of Lemma 5.3 we observe that, if K(J[2])/K is cyclic, then whether or not the discriminant of f (x) is a square in K can essentially be detected from the rational 2-torsion in J. In the statement, we continue to impose Convention 5.2.
Corollary 5.5. Suppose K (J[2]) /K is cyclic. Then ∆ f is a square in K if and only if one of the following holds: (i) (−1) dim J(K)[2] = 1 and either g is odd or f (x) has an odd degree factor over K, is the sign of σ as a permutation on the roots of f (x). Suppose σ has cycle type (d 1 , ..., d n ), so that the d i are the degrees of the irreducible factors of f (x) over K. Then we have . Moreover, K(J[2])/K contains at most one quadratic subextension, which yields a factorisation of f 0 (x) into 2 distinct conjugate polynomials if and only if each d i is even. The result now follows from Lemma 5.3.

Deficiency
Let K be a local field. Recall from Section 1.5.5 that a (smooth, proper, geometrically connected) curve X/K of genus g is said to be deficient if X has no K-rational divisor of degree g − 1 . In this section we collect some results on deficiency which will be of use later. Firstly, we determine the behaviour of deficiency in field extensions. Next, we give some criteria for determining when a hyperelliptic curve is deficient, which apply in particular when K(J[2])/K is cyclic. Finally, for nonarchimedean base fields we recall a criteria due to Poonen and Stoll which describes deficiency of a general curve in terms of its minimal proper regular model.
The first 2 results mentioned above are a consequence of the following description of deficiency, which arises as part of the proof of [PS99,Theorem 11]. Consider the short exact sequences of G K -modules Here K s (X) is the function field of X over the separable closure of K, Div(X K s ) is the group of divisors on the base-change of X to K s , Pic(X K s ) is the Picard group of X K s , and the map div sends a rational function on X K s to its associated divisor. As explained in the proof of [PS99, Theorem 11], combining the associated long exact sequences for Galois cohomology we obtain an exact sequence where Br(K) = H 2 (K, K s× ) denotes the Brauer group of K. Notation 6.3. We denote by φ K the composition where the first map is the one constructed above, and the second is the local invariant map.
By a result of Lichtenbaum [Lic69] (see also [PS99,Section 4]) X has a K-rational divisor class of degree g −1. Fix such a class L ∈ Pic g−1 (X K s ) G K . In the proof of [PS99, Theorem 11], Poonen-Stoll show that (g − 1)φ K (L) ∈ {0, 1/2}, and that X is deficient over K if and only if (g − 1)φ K (L) = 1/2. 6.1. Deficiency in field extensions. Recall that ǫ(X/K) ∈ {0, 1} is defined to be equal to 1 if X is deficient over K, and equal to 0 otherwise. Lemma 6.4. For any finite extension L/K we have Then L also gives a rational divisor class of degree g − 1 in Pic(X K s ) G L , and commutativity of the diagram 6.2. Deficiency for hyperelliptic curves. Now suppose that the characteristic of K is different from 2. Take C/K to be a hyperelliptic curve of genus g and fix a Weierstrass equation y 2 = f (x) for C. Since C has K-rational divisors of degree 2 (arising as the pull-back of rational points on P 1 K ), if g is odd then C is not deficient. Consequently, we impose the following assumption. Assumption 6.6. For the rest of this subsection, suppose that g is even.
Again using that C has K-rational divisors of degree 2, we see under this assumption that having a K-rational divisor of degree g − 1 is equivalent to having a K-rational divisor of any odd degree, which is in turn equivalent to having a rational point over some odd degree extension of K. In particular, if f (x) has an odd degree factor over K then C is not deficient.
The following proposition gives a convenient criterion for testing deficiency in the special case that f 0 (x) factors as a product of 2 conjugate polynomials over some quadratic extension of K.
Proposition 6.7. Suppose f 0 (x) factors over a quadratic extension F/K as a product of 2 polynomials conjugate under the action of Gal(F/K). Then C is deficient if and only if (c f , F/K) = −1.
and f 0b (x) are monic and conjugate under the action of Gal(F/K). As g is assumed even, both f 0a (x) and f 0b (x) necessarily have odd degree Denote by χ : G K → {±1} the quadratic character corresponding to F/K. Then for all σ ∈ G K we have , is invariant under G K . Further, we see that under the connecting map Pic(C K s ) G K → H 1 (K, K s (C) × /K s× ) associated to (6.1), the class [D a ] maps to the class of the 1-cocycle ρ defined by This lifts via the same formula to a 1-cochain valued in K s (C) × . The image of ρ under the connecting map Under inv K : Br(K) → Q/Z, the class of this 2-cocycle is mapped to 0 if c f is a norm from F × , and to 1/2 otherwise. Thus (g + 1)φ([D a ]) = 1/2 if and only if (c f , F/K) = −1.
Remark 6.8. As in Section 5, let W denote the G K -set of ramification points of the x-coordinate morphism C → P 1 K . Further, let T denote the quotient of the sum-1 part of the permutation module F W 2 by the diagonal action of F 2 . We see from (5.1) that T is a torsor under J[2]. In fact, T can be identified as a G K -set with the collection of theta characteristic on C K s (see [Mum71,Section 4] and recall that g is assumed even). From this we see that C has a K-rational theta characteristic if and only if either f (x) has an odd degree factor over K, or f 0 (x) factors as a product of 2 conjugate polynomials over some quadratic extension of K. Thus the results above concerning deficiency apply precisely when C has a K-rational theta characteristic.
Corollary 6.9. Suppose that K(J[2])/K is cyclic. If C is deficient over K then (g is even and) every irreducible factor of f (x) over K has even degree. When this is the case, denote by F/K the unique quadratic subextension of K(J[2])/K. Then C is deficient if and only if (c f , F/K) = −1.
Proof. As noted above we may assume that each irreducible factor of f (x) over K has even degree, in which case f (x) has degree 2g + 2. The assumption that K(J[2])/K is cyclic now ensures that there is indeed a unique quadratic subextension of K(J[2])/K, F/K say, and that f 0 (x) factors into 2 conjugate polynomials over F . The result now follows from Proposition 6.7.
Remark 6.10. Suppose we are in the situation of Corollary 6.9, so that (g is even and) K( Then the quadratic twist of C by L/K has Weierstrass equation C L : y 2 = df (x). As above, we have ǫ(C/K) = 0 = ǫ(C L /K) unless every irreducible factor of f (x) over K has even degree. In this latter case, with F/K as in the statement of Corollary 6.9, we see from that result that 6.3. Deficiency in terms of the minimal proper regular model. Now suppose that K is a nonarchimedean local field, possibly of characteristic 2, and that X/K is a (not necessarily hyperelliptic) curve of genus g. To conclude the section we recall a characterisation of deficiency in terms of the minimal proper regular model of X. We will make extensive use of this criterion later.
Let X /O K denote the minimal proper regular model of X, and let Xk denote the base-change tō k of its special fibre. Let {Γ i } i∈I denote the set of irreducible components of Xk. For each i ∈ I, let d i denote the multiplicity of Γ i in Xk, and let Orb G k (Γ i ) denote the G k -orbit of Γ i .
Lemma 6.11. The curve X is deficient over K if and only if Proof. This is observed by Poonen-Stoll; see [PS99], remark following the proof of Lemma 16.
Remark 6.12. When X/K is a hyperelliptic curve we have gcd i∈I {d i · |Orb G k (Γ i )|} ∈ {1, 2}. This follows from [BL99, Corollary 1.5] and the fact that all hyperelliptic curves have closed points of degree dividing 2.
7. First cases of Conjecture 1.7 In this section we prove Conjecture 1.7 in two cases: when K is archimedean, and when K has odd residue characteristic and J/K has good reduction. It will turn out that these are the only cases needed to prove Theorem 1.10 (in fact, even the case of archimedean K is not necessary for this). 7.1. Archimedean local fields. Here we consider Conjecture 1.7 for archimedean local fields. Clearly the only case of interest is the extension C/R. Let C/R be a hyperelliptic curve of genus g and let J/R be its Jacobian.
Proposition 7.1. Conjecture 1.7 holds for C/R and the extension C/R.

Proof.
We have w(J/C) = (−1) g (see, for example, [Sab07, Lemma 2.1]). Further, by [Mor19, Lemma 10.9 (ii)] we have |J(R)/N C/R J(C)| = 2 −g |J(R)[2]|. Denote by J −1 the quadratic twist of J by C/R, and and denote by C −1 the quadratic twist of C by C/R similarly. To verify Conjecture 1.7 we must show that = (∆ C , C/R), except when g is even and all irreducible factors of f (x) over R have even degree. In this latter case the two expressions differ by a sign. Since by Corollary 6.9 (cf. also Remark 6.10) this is exactly the case where ǫ(C/K) + ǫ(C −1 /K) = 1, we have the result. 7.2. Good reduction in odd residue characteristic. Suppose now that L/K is a quadratic extension of nonarchimedean local fields of odd residue characteristic. Let C/K be a hyperelliptic curve and J/K its Jacobian. We denote by v the normalised valuation on K.
Proposition 7.2. Suppose that J has good reduction over K. Then Conjecture 1.7 holds for C and the extension L/K.
Proof. Since J has good reduction over K we have w(J/L) = 1, so we are reduced to showing that . Moreover, the assumption on the reduction of J implies that K(J[2])/K is unramified. Thus adjoining a square root of ∆ C to K produces an unramified extension. In particular, v(∆ C ) is even and (∆ C , L/K) = 1. Finally, Corollary 6.9 gives (−1) ǫ(C/K)+ǫ(C L /K) = 1 also. Now suppose L/K is ramified. This time, [Mor19, Lemma 10.9 (i)] gives Moreover, as v(∆ C ) is even we have (∆ C , L/K) = 1 if and only if ∆ C is a square in K. We now conclude by Corollary 5.5 and Corollary 6.9.
8. Global conjectures imply instances of Conjecture 1.7 We have already proven enough cases of Conjecture 1.7 to prove Theorem 1.10.
Theorem 8.1 (=Theorem 1.10). Let K be a number field, C/K a hyperelliptic curve, J/K its Jacobian and v 0 a place of K. If the 2-parity conjecture holds for J over every quadratic extension F/K, then Conjecture 1.7 holds for J/K v 0 and every quadratic extension L/K v 0 .
Proof. Let L/K v 0 be a quadratic extension, and write L = K v 0 ( √ α). Let S be a finite set of places of K containing all places where J has bad reduction, all places dividing 2, and all archimedean places. Set T = S − {v 0 }. Now let F/K be a quadratic extension such that: Explicitly, we may take F = K( √ β) where β ∈ K is chosen, by weak approximation, to be sufficiently close to α v 0 -adically, and sufficiently close to 1 v-adically for all v ∈ T .
With such an extension F/K chosen, for a place v of K which is non-split in F/K, denote by v the unique place of F extending v. Then (cf. Theorem 2.1) the products are equal to w(J/F ) and (−1) rk 2 (J/F ) respectively, hence agree under the assumption that the 2parity conjecture holds for J over F .
On the other hand, by Proposition 7.2 and our assumptions on F/K, the individual contributions to these products at a place v agree, save possibly at v = v 0 . Thus the contributions at v = v 0 must agree also.

Unramified extensions
Let K be a nonarchimedean local field of characteristic different from 2. In this section we begin the study of Conjecture 1.7 for unramified quadratic extensions. Thus we fix a hyperelliptic curve C/K, and denote by L/K the unique unramified quadratic extension of K. As usual, we denote by J/K the Jacobian of C. Across Sections 9 and 10 we will prove the following: Proposition 9.1. Conjecture 1.7 for C/K and the extension L/K holds in each of the following cases: (i) the residue characteristic of K is odd, (ii) the residue characteristic of K is 2 and J/K has good reduction, (iii) the residue characteristic of K is 2, C has genus 2, and J/K has semistable reduction.
To prove this, the key fact we will exploit is that the formation of Néron models and minimal regular models commutes with unramified base-change. As L/K is assumed unramified this makes the quantities appearing in Conjecture 1.7 comparatively easy to describe and relate to one another (in particular, it allows us to readily relate invariants of C to invariants of the quadratic twist of C by L/K). This enables us to reduce Conjecture 1.7 to a statement which depends only on the curve C considered over the maximal unramified extension of K; see Corollary 9.9. We then prove this statement under the conditions of Proposition 9.1.
Denote by k the residue field of K, and denote by k L the residue field of L. Further, denote by f(J/K) the conductor of J, and denote by Φ the component group of the special fibre of the Néron model of J. Lemma 9.2 describes two of the terms appearing in Conjecture 1.7. Moreover, as L/K is unramified we have where v denotes the normalised valuation on K. We thus see that Conjecture 1.7 for C and L/K is equivalent to the assertion Since f(J/K) and v are unchanged under unramified extension, this predicts that the quantity is unchanged upon replacing K by a finite unramified extension F , and replacing L by the unique quadratic unramified extension F ′ /F . In fact, we can use this observation to predict a simpler expression for (9.5). We begin with some notation.
Notation 9.6. Denote by C the minimal regular model of C over O K . For each irreducible component Γ of Ck, write d(Γ) for its multiplicity. Further, denote by ι the automorphism of Ck induced from the hyperelliptic involution on C (which extends to an automorphim of C by uniqueness of the minimal regular model). We then define where here orb ι (Γ) denotes the orbit of an irreducible component Γ under the action of ι.
Now suppose that F/K is a sufficiently large even-degree unramified extension so that G k F acts trivially on both Φ(k) and on the set of irreducible components of Ck. Let F ′ /F be the unique quadratic unramified extension. Then we have , ǫ(C/F ) = 0 and ǫ(C F ′ /F ) = η(C). The first equality follows from our assumptions on the G k F -action on Φ(k), along with the description of the cohomology of cyclic groups given in [AW67, Section 8]. The second equality follows from Lemma 6.4 since F/K is assumed to have even degree. The third equality follows from Lemma 6.11, our assumptions on the G k F -action on the irreducible components of Ck, and the fact that the formation of the minimal regular model commutes with unramified base-change. Indeed, this last fact allows us to identify the geometric special fibre of the minimal regular model of C L /K with that of C/K, save with G k -action twisted by ι.
From (9.7) we find . Consequently, the discussion preceding Notation 9.6 predicts the following identity, which we will give an unconditional proof of.
Proposition 9.8. With the notation above, we have The proof of Proposition 9.8 that we will give is somewhat lengthy and we postpone it to the next section.
An immediate corollary of Proposition 9.8 is the following: Corollary 9.9. Conjecture 1.7 holds for C/K and the extension L/K if and only if (9.10) Remark 9.11. In the statement of Proposition 9.8 it is not simply true that , as the following example shows.
Example 9.12. Consider the genus 2 hyperelliptic curve where i is a square root of −1. Reducing the defining equation modulo 3 gives a semistable curve. In fact, the above equation viewed as a scheme over Z 3 , along with the usual chart at infinity, gives the stable model of C. After base-changing to Z nr 3 its special fibre consists of 2 rational curves, intersecting transversally at the 3 points (0, 0), (i, 0) and (−i, 0). The final two intersection points are swapped by the Frobenius element F ∈ G k , whilst the hyperelliptic involution ι : (x, y) → (x, −y) fixes each intersection point but swaps the 2 components. The minimal proper regular model of C over Z nr 3 is obtained by blowing up once each at the intersection points (i, 0) and (−i, 0). Its special fibre, along with the actions of F and ι, is thus as pictured: F ι Here each component pictured is a rational curve of multiplicity 1. Write L = Q 3 (i) for the unique unramified quadratic extension of Q 3 . Since F fixes the 2 components drawn horizontally we see from Lemma 6.11 that ǫ(C/Q 3 ) = 0. Similarly, we have η(C) = 0 since ι fixes the 2 components drawn vertically. However, each ι • F -orbit of components has size 2, so appealing to Lemma 6.11 once more we find ǫ(C L /Q 3 ) = 1. Thus However, one can also show (e.g. using the description of Φ(k) detailed later in Section 10.1) that Φ(k) ∼ = Z/8Z with F acting as multiplication by 5. Thus we have In particular, Proposition 9.8 holds in this example, even though neither of the individual congruences in Remark 9.11 hold. 9.1. Establishing (9.10) in odd residue characteristic. Assume that the residue characteristic of K is odd. Under this assumption we now establish the congruence (9.10).
Lemma 9.13. We have Proof. This is observed by Česnavičius in [Čes18, Lemma 4.2] (we remark that the cited result uses the assumption that K has odd residue characteristic).
In light of Lemma 9.13, to establish (9.10) it remains to show that is a squarefree polynomial of (without loss of generality) even degree 2g + 2. Let E/K nr be the field extension E = K nr (J[2]), and set G = Gal(E/K nr ). As explained in Section 5, E coincides with the splitting field of f (x) over K nr (recall that we are assuming g ≥ 2). Let G = G 0 ⊲ G 1 ⊲ G 2 ⊲ ... be the ramification filtration on G, and write g i = |G i |. Thus G 1 is the wild inertia group of E/K nr and is a p-group where p = char(k) (so in particular has odd order), and G/G 1 is cyclic.
to be equal to 1 if the genus g of C is even and each irreducible factor of f (x) over K nr has even degree. Otherwise, set η(f ) = 0.
Lemma 9.16. Let V = C[R] be the complex permutation representation of G associated to the set of roots of f (x). Then we have Proof. This will follow from the definitions of f(J[2]) and f(V ), along with a comparison between First let i ≥ 1 so that G i has odd order. Then necessarily f (x) has an odd degree factor over It remains to show that When g is even, Lemma 5.3 gives dim F 2 J[2] G = dim C V G − 2 + η(f ) and we are done. So suppose that g is odd. If f (x) has an odd degree factor over K nr then again we conclude immediately from Lemma 5.3. Finally, suppose each irreducible factor of f (x) over K nr has even degree. Write is monic. Applying Lemma 5.3 once again it suffices to show that there is a unique quadratic subextension of E/K nr over which f 0 (x) factors into 2 distinct, conjugate polynomials. To see this, first note that there is a unique quadratic subextension of E/K nr . Indeed, any such extension must necessarily be contained in E G 1 , and E G 1 /K nr is cyclic and (by the assumption on the degrees of the irreducible factors of f (x) over K nr ) has even degree. To see that f 0 (x) admits the required factorisation over this extension, let S = {h 1 , ..., h l } be the set of irreducible factors of f 0 (x) over E G 1 , each of which necessarily has odd degree. The cyclic group G/G 1 acts on S and since each factor of f 0 (x) over K nr has even degree, each orbit of G/G 1 on S has even order. Denote these disjoint orbits by S 1 , ..., S t , and for Fix a generator σ of G/G 1 and assume without loss of generality that is fixed by σ 2 , has σh(x) = h(x), and is such that f 0 (x) = h(x) · σh(x).

Next, we relate the Artin conductor of
Lemma 9.17. Let V = C[R] be as above. Then Res(h 1 , h 2 ) denotes the resultant of h 1 , h 2 ), we see that the discriminant of f (x) is, up to squares in K, the product of the discriminants of the f i (x). In this way we reduce to the case where f (x) is irreducible, which we treat now.
Assuming that f (x) is irreducible, let F/K be the splitting field of f (x) and let H be the stabiliser in Gal(F/K) of a root r ∈ R. Then V ∼ = C[Gal(F/K)/H] so by the conductor-discriminant formula [Ser79, VI.2 corollary to Proposition 4] we have f Combined, Lemmas 9.16 and 9.17 establish the congruence . To prove (9.10) it remains to reinterpret the 'correction' term η(f ).
Lemma 9.19. Let F/K be a finite unramified extension and denote by F ′ /F the unique quadratic unramified extension of F . Then, provided F/K is sufficiently large, we have η(f ) = ǫ(C F ′ /F ). In particular, we have η(f ) = η(C).
Proof. Arguing as in the proof of Lemma 9.16 we see that f (x) either has an odd degree factor over K nr , or factors as a product of 2 conjugate polynomials over the unique quadratic extension of K nr . From this we deduce that for every sufficiently large unramified extension F/K, either f (x) has an odd degree factor over F , or f (x) factors as a product of 2 conjugate polynomials over some totally ramified quadratic extension L/F (the latter happening if and only if each irreducible factor of f (x) over K nr has even degree). In the latter case, by enlarging F/K further if necessary we may also assume that the leading coefficient of f (x) is a norm from this quadratic extension. Since the quadratic twist C F ′ /F is given by the equation C F ′ : y 2 = uf (x) where u is a non-square unit in F , the claim that η(f ) = ǫ(C F ′ /F ) follows from Proposition 6.7 and the fact that (u, L/F ) = −1.
Corollary 9.20. Under the assumption that K has odd residue characteristic, (9.10) holds for C.
Proof. Lemma 9.19 allows us to replace η(f ) with η(C) in (9.18), hence establishing (9.14). Combining this with Lemma 9.13 gives the result. 9.2. Residue characteristic 2. We now give certain conditions under which the congruence (9.10) holds (or rather, can be shown to hold) when the residue characteristic of K is 2. Thus for the rest of this subsection we assume that K is a finite extension of Q 2 . 9.2.1. Good reduction of the Jacobian. If the Jacobian J/K of C has good reduction then we have (9.21) f(J/K) = 0 and Φ(k) = 0. Moreover, we have the following: Proposition 9.22. Under the assumption that J/K has good reduction, we have v(∆ C ) ≡ 0 (mod 2).
Proof. Let J /O K be the Néron model of J. The assumption that J has good reduction over K implies that J [2] is a finite flat group scheme over O K [Mil86,Proposition 20.7]. Letting e denote the absolute ramification index of K, it is a theorem of Fontaine that G u K acts trivially on J [2](K) = J[2] provided u > 2e − 1 [Fon85, Théorème A]. Note that we are using Serre's upper numbering for the higher ramification groups. Let F = K √ ∆ C and G = Gal(F/K), noting that F ⊆ K (J[2]). Combining the above discussion with Herbrand's theorem (see, for example, [Ser79, IV, Lemma 3.5]) we see that G u is trivial for u ≥ 2e. In particular, the conductor of F/K, which we denote f(F/K), satisfies f(F/K) ≤ 2e. Now suppose, for contradiction, that v(∆ C ) is odd. We thus have F = K ( √ π) for some uniformiser π of K. Letting σ denote the non-trivial element of G this gives where v F denotes the normalised valuation on F . From this we obtain f(F/K) = 2e+1, contradicting the bound above. Thus v(∆ C ) is even.
Remark 9.23. This proposition is trivially true also if J has good reduction and the residue characteristic of K is odd. Indeed, then J[2] is unramified so ∆ C is a square in K nr . In particular, ∆ C has even valuation.
Lemma 9.24. Under the assumption that J/K has good reduction, we have η(C) = 0.
Proof. Let C/O K denote the minimal regular model of C. Since J has good reduction, the curve C/K is semistable and the dual graph of Ck is a tree [Liu02, Proposition 10.1.51]. 2 Moreover, as there are no exceptional curves in Ck, each leaf corresponds to a positive genus component (which necessarily has multiplicity 1). Since the quotient of Ck by the hyperelliptic involution has arithmetic genus zero, the hyperelliptic involution necessarily fixes every leaf, hence acts trivially on the dual graph.
Corollary 9.25. If J/K has good reduction then (9.10) holds for C.
Proof. Combine (9.21) with Proposition 9.22 and Lemma 9.24. 9.2.2. Semistable curves of genus 2. When the genus of C is 2 we can draw on results of Liu [Liu94a] to establish additional cases of (9.10).
Proposition 9.26. Suppose that C/K is semistable and has genus 2. Then (9.10) holds for C.
Proof. This follows from Liu's genus 2 version of Ogg's formula [Liu94a, Theoreme 1]. Specifically, combining Theoreme 1, Theoreme 2 and Proposition 1 of [Liu94a], one obtains (independently of the residue characteristic of K) where n is the number of irreducible components of Ck (as usual C denotes the minimal regular model of C over O K ) and where d is defined in the statement of Liu's Theoreme 1. In [Liu94a, Section 5.2], Liu computes the term d−1 2 in a large number of cases, though not all in residue characteristic 2, in terms of the structure of Ck (more specifically, in terms of the 'type' of Ck as classified in [NU73] and [Ogg66]). This includes in particular all cases where C/K has semistable reduction. It is then easy to establish (9.10) for all semistable curves of genus 2 by combining this with the description, detailed in [Liu94b, Section 8], of the component group of a genus 2 curve in terms of its type.
9.3. Proof of Proposition 9.1. The above results combine to prove Proposition 9.1, conditional on the soon to be proven Proposition 9.8.
10. The proof of Proposition 9.8 Maintaining the setup of the previous section, we now turn to proving Proposition 9.8. We will deduce this from Theorem 10.2. This is a result applying to arbitrary curves, and which may be of independent interest. We begin by recalling a well-known description of the component group Φ(k) of the Jacobian of a (not necessarily hyperelliptic) curve in terms of its minimal regular model.
where D · Γ i is the intersection number between D and Γ i . Further, define β : Z I → Z by setting β(Γ i ) = d i and extending linearly. We have im(α) ⊆ ker(β). The natural action of G k on Xk makes Z I into a G k -module, and α is equivariant for this action. Endowing Z with trivial G k -action, the same is true of β. In this way, both im(α) and ker(β) become G k -modules.
Let J/K be the Jacobian of X, and denote by Φ its Néron component group. As explained in [BL99, Theorem 1.1], by work of Raynaud we have an exact sequence of G k -modules Denoting by F ∈ G k the Frobenius element, we thus have Note that F acts on Z I as a permutation of I, commuting with α and β, and preserving the arithmetic genus of components. We recall also from Lemma 6.11 that X is deficient over K if and only if 10.2. The result for general curves: statement. Maintaining the notation of the previous subsection, denote by G the group of all permutations of I commuting with the maps α and β and preserving the arithmetic genus of components. By assumption, G acts on im(α) and ker(β). Via the sequence (10.1) we have an induced action of G on Φ(k). For σ ∈ G we define In particular, viewing the Frobenius F ∈ G k as an element of G, we have q(F ) = 2 ǫ(X/K) . We will obtain Proposition 9.8 as a consequence of the following: Theorem 10.2. Let G be as above. Then the map D : Before proving Theorem 10.2, we explain how to use it to deduce Proposition 9.8.
Proof of Proposition 9.8 (conditional on Theorem 10.2). Maintaining the notation above, suppose X = C is a hyperelliptic curve over K. The hyperelliptic involution ι of C extends to an automorphism of the minimal regular model of C, and may therefore be viewed as an element of G. Moreover, as the induced automorphism ι * of the Jacobian of C is multiplication by −1, the action on Φ induced by ι is multiplication by −1 also (cf. proof of [BL99, Theorem 1.1]). Thus Let L denote the unique quadratic unramified extension of K and, as usual, denote by F the Frobenius element in G k . Then we have Here for the first equality we are using the description of the cohomology of cyclic groups given in [AW67, Section 8]. From Lemma 6.4 we have ǫ(C/L) = 0, hence q(F 2 ) = 1. Moreover, we have η(C) = ord 2 q(ι), ǫ(C/K) = ord 2 q(F ), and ǫ(C L /K) = ord 2 q(ι • F ). To obtain the last equality we note that, since the formation of minimal regular models commutes with unramified base-change, we may identify the geometric special fibre of the minimal regular model of C L /K with that of C/K, save with G k -action twisted by the hyperelliptic involution.
On the other hand, it follows from Theorem 10.2 that as elements of Q × /Q ×2 . Taking 2-adic valuations of this equation, and interpreting the resulting terms via the discussion above, we obtain the congruence of Proposition 9.8.

Proof of Theorem 10.2.
In what follows we take the notation introduced in the statement of Theorem 10.2.
Lemma 10.3. For each σ ∈ G we have (That is, as elements of Q × /Q ×2 , D(σ) is the absolute value of the determinant of α viewed as an endomorphism of (σ − 1)Q I .) We remark that, in order to apply the cited result, we are using the assumption that G preserves the arithmetic genus of components.
In what follows, to ease notation we write Λ for the Z[G]-module Z I . Now We now apply the snake lemma to the commutative diagram with exact rows where each vertical arrow is induced by β. Noting that all vertical arrows are surjective, this gives Thus as a function G → Q × /Q ×2 we have where for the last equality we use that σ commutes with α.
To conclude, note that (σ − 1)Λ is a free Z-module of finite rank, and α is a linear endomorphism of this group. By properties of Smith normal form, the order of the group is equal to the absolute value of the determinant of α as an endomorphism of the Q-vector space (σ − 1)Q I . This gives the result.
The passage from Z[G]-modules to Q[G]-modules provided by Lemma 10.3 allows us to make use of representation theory in characteristic zero. Note that the matrix representing α on Q I with respect to the natural permutation basis is symmetric (it's just the intersection matrix associated to Xk). We thus see that the minimal polynomial of α as an endomorphism of Q I splits over R.
Moreover, the kernel of α is ( i∈I d i Γ i ) · Q, which is fixed by G. These observations motivate (and allow us to apply) the following lemmas.
Lemma 10.4. Let G be a finite cyclic group with generator σ, and let V be a Q[G]-representation. Let α ∈ End Q[G] V be a G-endomorphism of V whose minimal polynomial splits over R and is such that ker(α) ⊆ V G . Then Suppose, without loss of generality, that V 1 is the trivial representation. Now α preserves this decomposition and (σ − 1)V = n i=2 V d i i . Since ker(α) ⊆ V G , the restriction of α to each V d i i is non-singular. Thus we reduce to the case where α is non-singular and V = W d for an irreducible non-trivial Q[G]-representation W . Let χ be the character of a complex irreducible constituent of W . We can suppose that χ is non-real (so that χ(σ) / ∈ {±1}), in which case we wish to show that det W so that D is a division algebra, and let K/Q be the centre of D. We have K ∼ = Q(χ) as Q-algebras, where Q(χ) is the character field of χ. Se, for example, [Rei61] for proofs of the representation theoretic facts used above. Note that K/Q is abelian. As χ is assumed non-real, the field K in not totally real hence there is an index 2 totally real subfield K + of K (recall that K/Q is abelian). We claim that det(α) is in K + . Indeed, since the minimal polynomial of α as a Q-endomorphism of V splits over R, each root of the minimal polynomial of α as a K-endomorphism of V is totally real. It follows that det(α) is a product of totally real numbers, hence in K + . Thus as desired.
Lemma 10.5. Let G be a finite group and V a Q[G]-representation. Let α ∈ End Q[G] V be a Gendomorphism of V whose minimal polynomial splits over R and is such that ker(α) ⊆ V G . Then the function φ : G → Q × /Q ×2 defined by Once again we have K ∼ = Q(χ), K/Q is abelian, and we may view α as a K[G]-endomorphism of V . Moreover, we similarly have det (α|V −1,σ ) = N K/Q (det K (α|V −1,σ )) for all σ ∈ G. Denoting by φ K the function G → K × /K ×2 given by First suppose that K in not totally real. Then there is an index 2 totally real subfield K + of K. Since the minimal polynomial of α as a Q-endomorphism of V splits over R, det (α|V −1,σ ) lies in hence φ is trivial in this case. Now assume that K is totally real, or equivalently that χ is real valued. Let m be the Schur index of χ (over Q or equivalently K). Suppose first that χ is realisable over R. Then, via a chosen embedding K ֒→ R, we have V ⊗ K R ∼ = U md for some irreducible real representation U . Fix σ ∈ G.
The determinant of α, viewed as a K-endomorphism of (U −1,σ ) md , is then equal to det(M ) dimU −1,σ . In fact, one sees that det(M ) is equal to Nrd(α) ∈ K × where here Nrd denotes the reduced norm on the central simple algebra We claim that the congruence (10.6) dimU −1,σ + dimU −1,τ ≡ dimU −1,στ (mod 2) holds for all σ and τ in G. Combined with the above discussion this shows that φ K , hence φ, is a homomorphism in this case. To prove the claim we note that U is a real vector space and each σ ∈ G acts on U as a finite order matrix N σ which is hence diagonalisable over C. Basechanging to C, diagonalising N σ , and noting that the eigenvalues of N σ are roots of unity appearing in complex-conjugate pairs, one sees that for each σ ∈ G we have The congruence (10.6) now follows from multiplicativity of the determinant. Finally, suppose that χ is not realisable over R. Then we have V ⊗ K C ∼ = U md where U is an irreducible representation over C such that U , hence U md , possesses a non-degenerate G-invariant alternating form. Denote by , such a form on U md . The argument for the previous case again gives det K (α|V −1,σ ) = Nrd(α) dimU −1,σ . We claim that dimU −1,σ is even for each σ ∈ G, from which it follows that φ K , hence φ, is trivial. Indeed, the pairing , gives a G-equivariant isomorphism from U to its dual U * . This isomorphism respects the σ-eigenspace decomposition on each side and hence restricts to an isomorphism U −1,σ ∼ −→ U * −1,σ whose associated bilinear pairing is non-degenerate and alternating. Thus dim U −1,σ is even.
Proof of Theorem 10.2. In the notation of Section 10.1, let V denote the Q[G]-representation Q I . For σ ∈ G, combining Lemma 10.3 with Lemma 10.4 we see that we have (That the assumptions of Lemma 10.4 are satisfied follows from the discussion preceding the statement of that lemma.) The result now follows from Lemma 10.5. 10.5. Computing Tamagawa numbers modulo squares. The proof of Theorem 10.2 facilitates the computation of Tamagawa numbers of hyperelliptic curves, at least up to squares, and to end the section we record this in the following proposition. To make the statement more self-contained we summarise now the notation used.
Notation 10.7. We take the following notation: • K a non-archimedean local field, • X/K a smooth, proper, geometrically connected curve of genus g, • Φ the Néron component group (scheme) of the Jacobian of X, • X /O K the minimal proper regular model of X, • Xk the special fibre of X base-changed tok, • Γ 1 , ..., Γ n the irreducible components of Xk, and r i the size of the G k -orbit of Γ i , • ǫ(X/K) ∈ {0, 1}, defined to be equal to 1 if X is deficient over K, and 0 otherwise, • F the Frobenius element in G k .
Proposition 10.8. Take the notation above. Moreover, let S 1 , ..., S t be the even sized orbits of G k on the set {Γ 1 , ..., Γ n } of irreducible components. For each 1 ≤ i ≤ t let m i = |S i |, let Γ i,1 be a representative of the orbit S i , and define .
where ·, · denotes the intersection pairing on Xk.

Ramified extensions in odd residue characteristic: generalities
Let K be a nonarchimedean local field of odd residue characteristic. In this section we consider Conjecture 1.7 when the quadratic extension L/K is ramified. Specifically, across Sections 11 to 15 we will prove the following: Proposition 11.1 (=Proposition 15.1). Let C/K be a hyperelliptic curve and let L/K be a ramified quadratic extension. If C/K has semistable reduction, then Conjecture 1.7 holds for C and the extension L/K.
Thus for this section we fix a ramified quadratic extension L/K, and fix a hyperelliptic curve C/K with semistable reduction. As usual we denote by J the Jacobian of C. Recall from Lemma 3.4 that, since K has odd residue characteristic, we can express the cokernel of the local norm map in terms of Tamagawa numbers: We begin by describing a method for computing the ratio c(J/L) c(J/K) up to rational squares for general semistable curves. Separately, we will compute c(J L /K) up to squares by analysing the minimal regular model of the quadratic twist of C by L. Since C L /K is not semistable, we use results from Section 10 to facilitate this computation. As we shall see, the terms in Conjecture 1.7 involving deficiency and root numbers will naturally appear along the way. For a more precise description of the strategy for proving Proposition 11.1, see Section 11.3 below.
11.1. The minimal proper regular model of a semistable curve. For proofs and more details of what follows we refer to [SGA7 I ]. For the specific formulation detailed below we refer to [DDMM18, Section 2] and the references therein.
Denote by C/O K the minimal proper regular model of C, and denote by Ck the special fibre of C, base-changed tok. Since C/K is assumed semistable, Ck is a semistable curve overk. Let Υ C denote the dual graph Ck; by definition this is the finite connected graph with a vertex for each irreducible component of Ck, and such that vertices corresponding to components Γ 1 and Γ 2 are joined by one edge for each ordinary double point of Ck lying on both Γ 1 and Γ 2 (in particular Υ C may have loops and multiple edges). We view Υ C as a metric space where we give each edge length 1. Denote by H 1 (Υ C , Z) the first singular homology group of Υ C . Since Ck is the base-change from k tok of the special fibre of C, H 1 (Υ C , Z) carries a natural G k -action. 3 Moreover, H 1 (Υ C , Z) carries a natural non-degenerate, symmetric, G k -invariant bilinear pairing P : H 1 (Υ C , Z) × H 1 (Υ C , Z) → Z (informally, P (γ, γ ′ ) is the signed length of γ ∩ γ ′ ). The pairing P induces an injection sending γ to P (−, γ). The component group Φ(k) of J/K is then G k -equivariantly isomorphic to the cokernel of this map: In particular, we have Moreover, the root number w(J/K) of J is encoded in the G k -invariants of H 1 (Υ C , Z): If we replace K by L and repeat the above constructions for the base-change C L of C to L, then the dual graph Υ C L is obtained from Υ C by replacing each edge by a path consisting of 2 edges. 4 In particular, the homology of the new dual graph with its G k -action is unchanged, but the pairing gets multiplied by 2. Thus we have 11.2. The group B C/K . Following work of Betts-Dokchitser [BD19], the quantities appearing in (11.5) and (11.6) can be neatly packaged together in the following way. Temporarily writing Λ = H 1 (Υ C , Z), define the finite abelian group B C/K by (11.7) where the map is induced by (11.3). Combining (11.5) and (11.6) with [BD19, Theorem 1.4.2] then gives Remark 11.9. If T denotes the toric part of the Raynaud parametrisation of J/K, and X(T ) denotes its character group, then X(T ) carries a natural G k -action and a non-degenerate symmetric pairing . As explained in [DDMM18, Section 2], it follows from work of Raynaud that X(T ) ∼ = H 1 (G k , Z) as G k -modules with a pairing. One can then alternatively obtain (11.8) directly from [BD19, Theorem 1.1.1 (i)].
11.3. Strategy for the proof of Proposition 11.1. In light of (11.2) and (11.8), Conjecture 1.7 for C and L/K is the equivalent to the assertion (11.10) (−1) dim B C/K [2] ? = (−1) ord 2 c(J L /K) (∆ C , L/K)(−1) ǫ(C/K)+ǫ(C L /K) . In Section 12 below we give a general result, Proposition 12.18, which facilitates the computation of the parity of the dimension of the 2-torsion of the group B Λ := im H 1 G k , Λ −→ H 1 G k , Λ ∨ associated to an arbitrary G k -lattice Λ equipped with a non-degenerate symmetric bilinear pairing.
In Section 13 we summarize results from [DDMM18] which give an explicit description of the lattice H 1 (Υ C , Z) attached to a semistable hyperelliptic curve C/K : y 2 = f (x) in terms of combinatorial data associated to the p-adic distances between the roots of f (x). Combined with the results of Section 12 mentioned above, this enables the explicit computation of dim B C/K [2] (mod 2) for arbitrary semistable hyperelliptic curves; we present the result of this computation as Corollary 13.25. (Strictly speaking, we only carry out these computations over a suitably large odd degree unramified extension of K. This suffices for the application to Conjecture 1.7 thanks to Lemma 4.1, and has the advantage that several statements in [DDMM18] simplify after such an extension.) Separately, in Section 14 we present an explicit combinatorial description of the minimal proper regular model of a ramified quadratic twist of a semistable hyperelliptic curve. This will be deduced from work of Faraggi-Nowell [FN20] which more generally describes the minimal regular SNC model of a hyperelliptic curve X over a local field of odd residue characteristic, under the assumption that X attains semistable reduction after a tamely ramified extension of the base field. We combine this description with Proposition 10.8 to describe explicitly the quantity (−1) ord 2 c(J L /K)+ǫ(C L /K) ; we present this result as Corollary 14.30.

The parity of dim B Λ [2]
The aim of this section is to prove Proposition 12.18, which gives an explicit criterion for determining the parity of dim B Λ [2], where B Λ is the group defined in Section 12.1 below and considered by Betts-Dokchitser in [BD19]. This can be viewed as a complement to the results of [BD19, Section 2].
Let k be a finite field. Take Λ to be a (discrete) Z[G k ]-module, free and of finite rank as a Z-module, and equipped with a non-degenerate G k -invariant symmetric bilinear pairing (12.1) , We extend , bilinearly to a pairing on the Q[G k ]-module V := Λ ⊗ Z Q and write Λ ∨ for the dual lattice The map v → v, − identifies Λ ∨ with Hom(Λ, Z). We denote by Φ the finite abelian group Λ ∨ /Λ, the discriminant group of the lattice. The pairing on V restricts to a G k -invariant pairing Λ ∨ × Λ → Z, and further induces a non-degenerate symmetric bilinear pairing 12.1. The group B Λ . Define the finite abelian group Consider the pairing Here the first map is composition of cup-product with the pairing (12.1), and the second is the inverse of the coboundary map δ : H 1 (G k , Q/Z) → H 2 (G k , Z) arising from the short exact sequence The final map is given by evaluating cocycles at the Frobenius element F ∈ G k . We have the following result of Betts-Dokchitser.   → (x, x). This is a homomorphism by antisymmetry of ( , ), so by non-degeneracy there is a unique c ∈ B Λ such that It follows from the arguments of [PS99, Section 6] that one has In Lemma 12.15 below we give an explicit description of this class c. The construction involves quadratic refinements of the pairings (12.1) and (12.3).
Notation 12.11. For abelian groups A and M , call a function q : A → M a quadratic map if the function B q : A × A → M defined by B q (a 1 , a 2 ) = q(a 1 + a 2 ) − q(a 1 ) − q(a 2 ) is bilinear. We call q a quadratic form if moreover, for all a ∈ A and n ∈ Z, we have q(na) = n 2 q(a). We say that q is a quadratic refinement of B q . Denote by Q Λ the set of Z-valued quadratic refinements of (12.1), and by Q Φ the set of Q/Z-valued quadratic refinements of (12.3).
Now define the subset S ⊆ V as One checks using the fact that λ → λ, λ (mod 2) is a homomorphism that S is nonempty. For v ∈ S denote by q v : Λ → Z the quadratic map (12.13) q v (λ) = 1 2 ( λ, λ + λ, v ) . Sending v to q v gives a bijection from S to Q Λ . Moreover, taking λ ∈ Λ ∨ in the formula (12.13), and then reducing modulo Z, gives a quadratic refinement q v : Φ → Q/Z of the pairing (12.3). This is a quadratic form if and only if v ∈ Λ. The map v → q v is a bijection between S/2Λ (the quotient of S by the action of 2Λ) and Q Φ . 12.3. Cohomology classes associated to quadratic refinements. Since the pairing (12.1) is G k -invariant, G k acts on Q Λ . Explicitly, for σ ∈ G k and q ∈ Q Λ we define σ q : Λ → Z by setting σ q(λ) = q(σ −1 λ). Given q 1 , q 2 ∈ Λ we have q 1 − q 2 ∈ Hom(Λ, Z) = Λ ∨ ; thus Λ ∨ acts simply transitively on Q Λ . In particular, associated to Q Λ is a class q Λ ∈ H 1 (G k , Λ ∨ ), explicitly represented by the 1-cocycle σ → σ q − q for any q ∈ Q Λ . We similarly have q Φ ∈ H 1 (G k , Φ) associated to Q Φ . 5 Remark 12.14. The discussion in Section 12.2 above provides a more explicit description of the classes q Λ and q Φ . Let v ∈ S and let q v (resp. q v ) be the associated element of Q Λ (resp. Q Φ ). Computing the associated cocycles one sees that q Λ and q Φ are represented by the 1-cocycles Lemma 12.15. With the notation above, we have the following.
Here both cup-products are induced by the pairing , . From this identity and commutativity of (12.7) it follows that q Λ ∪ ρ = 0 for all ρ ∈ ker H 1 (G k , Λ) → H 1 (G k , Λ ∨ ) . It now follows formally from the stated properties of the pairings in (12.7) that q Λ ∈ B Λ . This proves part (i), and part (ii) now follows from (12.16) and the definition of the pairing ( , ) on B Λ .
Remark 12.17. Since q Λ is a lift of q Φ to H 1 (G k , Λ ∨ ), it follows from Lemma 12.15 (i) that the class q Φ ∈ H 1 (G k , Φ) is trivial. In particular, the pairing (12.3) on Φ admits a G k -invariant quadratic refinement. From the discussion in Section 12.2, such a quadratic refinement is necessarily of the form q v for some v ∈ S. The G k -invariance of q v means that such a v satisfies σv − v ∈ 2Λ for all σ ∈ G k . 12.4. The order of B Λ modulo squares. We now give the promised criterion for determining the parity of dim B Λ [2]. As usual, F ∈ G k denotes the Frobenius element. The set S is as defined in (12.12).
Proposition 12.18. There exists x ∈ S such that 1 . Proof. For existence, take any x ∈ S for which the associated quadratic form q x is G k -invariant (cf. Remark 12.17). Now fix such an x and denote by a : G k → Λ the 1-cocycle a(σ) = 1 2 (σx − x). Its class in To be more precise, mimicking the construction of qΛ yields a class in H 1 (G k , Φ * ) where Φ * = Hom(Φ, Q/Z); we transport this class to H 1 (G k , Φ) via the isomorphism Φ ∼ → Φ * provided by the pairing (12.3).
Here the coboundary maps δ arise from the short exact sequence (12.5) and the corresponding sequence given by tensoring (12.5) by Λ ∨ , and both cup-products are induced by the pairing , . The element 1 2 x ∈ V /Λ ∨ defines an element of H 0 (G k , V /Λ ∨ ) which maps under δ to q Λ . From commutativity of (12.19) and the definition of the pairing ( , ) on B Λ , we find The result now follows from (12.10) and Lemma 12.15 (ii).

Clusters and the group B C/K for semistable hyperelliptic curves
We take the notation of Section 11. In particular, K denotes a nonarchimedean local field with residue field k of odd characteristic, and C/K denotes a hyperelliptic curve with semistable reduction. We henceforth fix a Weierstrass equation is a squarefree polynomial of degree 2g + 1 or 2g + 2 for g ≥ 2 the genus of C. We denote by R the set of roots of f (x) in K s and denote by c f the leading coefficient of f (x). Thus 13.1. Clusters. We now recall several results from [DDMM18], which provides a framework for studying invariants of hyperelliptic curves over local fields of odd residue characteristic. We refer to that work for more details (cf. also [BBB + ]). The central object is that of a cluster. In what follows we denote by v :K → Q ∪ {∞} the extension toK of the normalised valuation on K.
Definition 13.1. A cluster is a non-empty subset s ⊂ R of the form s = D ∩ R for some disc D = {x ∈K | v(x − z) ≥ d} where z ∈K and d ∈ Q. If |s| > 1 then s is said to be proper and its depth d s is defined as d s = min r,r ′ ∈s v(r − r ′ ).
We call any element z s of the minimal disc cutting out a proper cluster s a centre for s.
We summarize some terminology for clusters.
Definition 13.2. Given clusters s 1 = s 2 with s 1 a maximal subcluster of s 2 , we say that s 1 is a child of s 2 , denote this s 1 < s 2 , and refer to s 2 as the parent of s 1 . Any cluster s = R has a unique parent P (s). We define the relative depth of a proper cluster s = R as δ s := d s − d P (s) ≥ 0. We call a cluster even (resp. odd) if it contains an even (resp. odd) number of roots, and call it übereven if it is even and all its children are even also. We call a cluster s principal if |s| ≥ 3, save when either s = R is even and has exactly two children, or when s has a child of size 2g. A cluster of size 2 is called a twin, and a non-übereven cluster that has a child of size 2g is called a cotwin. For a principal cluster s, its genus g(s) is defined as g(s) = 1 2 (#{odd children of s} − 1) .
Finally, for clusters s 1 , s 2 , we write s 1 ∧ s 2 for the smallest cluster containing both s 1 and s 2 .
Example 13.3. Take C to be the hyperelliptic cuve considered in Example 9.12, where i is a square root of −1. The proper clusters are The unique principal cluster is R and it is übereven, has depth 0 and genus 0. There are 3 twins: t 1 , t 2 and t 3 , and we have δ t 1 = 1/2, δ t 2 = δ t 3 = 1. We display this information pictorially as shown: Here we draw roots as and draw ovals around roots to represent a proper cluster. The subscript on the outer cluster is its depth, and on all other clusters it is the relative depth. We refer to this diagram as the cluster picture of C.
We remark that the description of the minimal regular model of C given previously in Example 9.12 now follows immediately from [DDMM18, Theorems 1.11 and 8.6].
For the rest of this section we make the following assumption, which will lead to several simplifications in the results of [DDMM18].
Assumption 13.4. We assume that |R| = 2g + 2 and that there are no clusters of size 2g or 2g + 1.
Remark 13.5. By [DDMM18, Theorem 15.2], any semistable hyperelliptic curve over K is isomorphic to a curve satisfying Assumption 13.4 over any suitably large odd degree unramified extension of K (the key point being that if the residue field of K is sufficiently large then one can make a change of variables to force Assumption 13.4 to be satisfied).
We now summarize certain results from [DDMM18], using Assumption 13.4 to simplify several statements. First, the fact that C is semistable forces several constraints on the possible clusters and their depths. Specifically we have: Theorem 13.6 ([DDMM18] Theorem 1.8). Semistability of C is equivalent to the following three conditions: (1) the extension K(R)/K has ramification degree at most 2, (2) every proper cluster is invariant under the action of the inertia group of K, (3) every principal cluster s has d s ∈ Z and ν s ∈ 2Z, where ν s is the quantity Note that by part (1) of Theorem 13.6, each inertia orbit of roots has size at most 2 (i.e. the irreducible factors of f (x) over K nr are linear or quadratic), and every cluster s has d s ∈ 1 2 Z. The set of proper cluster s with d s / ∈ Z will be of particular importance.
Notation 13.8. Let T denote the set of proper clusters s with d s / ∈ Z.
Lemma 13.9. If r = r ′ are inertia conjugate elements of R, then s = {r, r ′ } is a cluster with d s / ∈ Z. Moreover, every proper cluster s with d s / ∈ Z takes this form.
Proof. If r and r ′ are inertia conjugate roots then v(r − r ′ ) ∈ 1/2 + Z (cf. [DDMM18, Lemma C.2]), so the minimal cluster containing both r and r ′ has non-integer depth. In light of Assumption 13.4, it follows from [DDMM18, Lemma 4.2] that {r, r ′ } is a cluster. Moreover, [DDMM18,Lemma 4.2] shows further that any proper cluster s with d s / ∈ Z takes this form.
A consequence of Lemma 13.9 is that T is naturally in bijection with the set of inertia-conjugate pairs of roots of f (x).
13.2. Signs associated to clusters. By Theorem 13.6(2), the assumption that C is semistable means that Gal(K nr /K) = G k acts on the set of proper clusters. We will augment this action by adding in certain signs associated to even clusters (cf. [DDMM18, Definition 1.12]).
Notation 13.10. For a cluster s, we write s * for the smallest cluster s * ⊇ s whose parent is not übereven, and set s * = R if no such cluster exists.
Definition 13.11. For even clusters s fix a choice of θ s = c f r / ∈s (z s − r), where z s is any centre for s. Still assuming s is even, define ǫ s : Here m denotes the maximal ideal of the ring of integers ofK, so that 'mod m' denotes reduction to the residue fieldk.
Remark 13.12. If s = R is an even cluster, or s = R is übereven, then v c f r / ∈s (z s − r) = ν s − |s|d s is an even integer (here ν s is as defined in the statement of Theorem 13.6). Indeed, by [DDMM18, Lemma C.5] we have ν s − |s|d s = ν P (s) − |s|d P (s) , and by Lemmas 4.2 and 4.9 of op. cit. we see that one of s or P (s) must have both integral depth and even ν. In particular, it follows that θ s ∈ K nr . Thus ǫ s descends to a function G k → {±1}.
Remark 13.13. As explained in [DDMM18, Remark 1.14], although the function ǫ s depends on the choice of θ s , the restriction of ǫ s to the stabiliser of s does not. In fact, if K s denotes the fixed field ofK by the stabiliser of s, then K s is a finite unramified extension of K, θ 2 s * ∈ K s , and ǫ s restricted to the stabiliser of s is the quadratic character associated to the extension K s (θ s * )/K s .
Example 13.14. Let C be as in Example 13.3 and let s be any of the 4 even clusters. Then we have s * = R. We can take z R = 0 and θ R = 1. Then ǫ s (σ) = 1 for all σ.
We remark that the description of the Frobenius action on the special fibre of the minimal regular model of C detailed previously in Example 9.12 can be read off from this data, coupled with the Frobenius action on the set of proper clusters; see [DDMM18, Theorems 8.6].
13.3. Description of the lattice. We retain the notation from Section 11.1. In particular, Υ C denotes the dual graph of the (geometric special fibre of the) minimal proper regular model of C. Here we recall from [DDMM18] a description of the Z[G k ]-module H 1 (Υ C , Z) along with its associated pairing. It will be convenient to first define an auxiliary lattice Π which is closely related to H 1 (Υ C , Z), but which is simpler to describe.
Definition 13.15. Let A be the set of even non-übereven clusters excluding R. Define Further, let B be the subset of A consisting of clusters s with s * = R. We endow Π with the symmetric pairing , : Π × Π −→ Z given by s 1 , s 2 ∈ B. We further endow Π with the G k -action given by σ · ℓ s = ǫ s (σ)ℓ σs . Note that the pairing , is invariant for this action.
It will be useful to note that the pairing on Π/2Π induced from that on Π has a very simple form.
Proof. Combine Lemma 13.9 with the formula (13.16). 13.4. The group B C/K for semistable hyperelliptic curves. Let B C/K be the group defined in (11.7). Let Λ be as in Definition 13.18 above, and let B Λ be the associated group defined in Section 12.1. By Theorem 13.19 we have B C/K ∼ = B Λ . Recall from Notation 13.8 the definition of the set T . We denote by F ∈ G k the Frobenius element, and for a cluster s we denote by Orb s its G k -orbit.
We begin with the following lemma which will be needed during the proof.
Lemma 13.21. Suppose that B = ∅ and that all s ∈ B T have |Orb s | even. Then Proof. Note that the assumption that B is non-empty means that R is übereven. Let s = R be a proper cluster with s / ∈ B ∩ T . We claim that |Orb s | is even. Indeed, the assumptions on s mean in particular that s is contained in a child s ′ of R. Clearly s ′ cannot be in T . Thus s ′ ∈ B T , hence |Orb s ′ | is even by assumption. Since s ⊆ s ′ it follows that |Orb s | is even also, proving the claim. The assumption that B = ∅ means that either R is non-principal or g(R) = 0. Thus each principal cluster of positive genus has an even sized G k -orbit, so the second term on the right hand side of (13.22) is an even integer. We therefore have |A| − 1 = rkΛ ≡ g (mod 2).
By the initial claim, every cluster s ∈ A has |Orb s | even, save possibly for those clusters in B ∩ T . Thus |A| ≡ |B ∩ T | (mod 2) and the result follows.

Further, we have
It follows that F t−t ∈ 2Λ (note that if B = ∅ then for any s ∈ T ∩B we have ǫ F −1 s (F ) = ǫ R (F ) = 1). Taking x = t in Proposition 12.18 then gives the last congruence following from Lemma 13.17. 6 Thus we have the result in this case. Case 2: B = ∅, ǫ R (F ) = −1, |B ∩ T | even. Write B ∩ T = {s 1 , ..., s 2k }. This time we set t = s∈T B ℓ s + 2k i=1 (−1) i ℓ s i , noting that t ∈ Λ. As with the previous case, taking x = t in Proposition 12.18 gives the result (note that by Lemma 13.21 we are trying to show that dim B C/K [2] ≡ N (mod 2) in this case).
Case 3: B = ∅, ǫ R (F ) = −1, |B ∩ T | odd, |Orb s | odd for some s ∈ B T . Choose some s 1 ∈ B T with m 1 = |Orb s 1 | odd, and write m 2 = |B ∩ T |. This time, take which lies in Λ by construction. Again, we conclude by taking x = t in Proposition 12.18.
Case 4: B = ∅, ǫ R (F ) = −1, |B ∩ T | odd, all s ∈ B T have |Orb s | even. Note that in this case we have g even by Lemma 13.21, so we want to show that dim B C/K [2] ≡ N + 1 (mod 2). Since each s ∈ B T has |Orb s | even, we can partition B T into two disjoint sets B 0 and B 1 with F (B 0 ) = B 1 . For s ∈ A, write ℓ ∨ s for the element of Hom(Λ, Z) sending ℓ s to 1, and sending ℓ s ′ to 0 for each s ′ = s. Consider the element Then we have λ, λ ≡ φ(λ) (mod 2), as follows from Lemma 13.17 upon noting that Λ is by definition the collection of elements s n s ℓ s ∈ Π for which s∈B n s = 0. Moreover, we have Since s∈T B (ǫ F −1 s (F ) − 1)ℓ s ∈ 2Λ we can apply Remark 12.20 to φ, giving This latter quantity is congruent to N + 1 modulo 2. Indeed, every element of B ∩ T has ǫ s (F ) = −1 by assumption. Since moreover |B ∩ T | is assumed odd, the claimed congruence follows.
Remark 13.23. Instead of appealing to Proposition 12.18, an alternative approach to proving Proposition 13.20 might be to draw on work of Betts [Bet22, Section 3] (see also [BBB + , Section 10]), which gives a description in terms of clusters for the individual Tamagawa numbers c(J/L) and c(J/K). From this one might then hope to prove the result by computing explicitly the quotient c(J/L)/c(J/K) and appealing to (11.8). However, the description of Tamagawa numbers given in that work becomes sufficiently complicated in the presence of übereven clusters that we have elected to avoid this approach.
Before stating the final result of the section we require one further piece of notation.
Notation 13.24. Define κ(C) ∈ {0, 1} as follows. We set κ(C) = 1 if R = s 1 ⊔ s 2 is a disjoint union of 2 odd G k -conjugate clusters s 1 and s 2 with both δ s 1 and δ s 2 odd (note in particular that this forces C to have even genus). We set κ(C) = 0 otherwise.
Corollary 13.25. We have Proof. Combine Proposition 13.20 with [DDMM18, Theorem 1.23] (the cited result gives an explicit description of deficiency in terms of clusters; to apply it recall that we have a running assumption that R has no cotwins).
14. Ramified quadratic twists of semistable hyperelliptic curves We retain the notation and setup of the previous section. Thus K is a nonarchimedean local field of odd residue characteristic, and C/K : y 2 = f (x) is a semistable hyperelliptic curve over K. We continue to impose Assumption 13.4, so that f (x) has even degree and the set R of roots of f (x) in K has no cotwins in the sense of Definition 13.2. Let L/K be a ramified quadratic extension of K, and write L = K( √ π) for some uniformiser π ∈ K.
14.1. The minimal regular model of C L . We now give an explicit 'cluster picture' description of the special fibre of the minimal regular model of the quadratic twist C L : y 2 = πf (x) of C by L. As we shall see, even though C L is no longer semistable over K, one can still give a simple description of its minimal regular model in terms of clusters. To avoid confusion when comparing invariants of C with invariants of C L later we will make the following convention.
Convention 14.1. Unless stated otherwise, in this section we will view all clusters as being associated to the polynomial f (x) defining C, as opposed to the polynomial πf (x) defining C L . Since both the clusters themselves and the associated functions d s and δ s (depth and relative depth) are functions purely of the set of roots R, they are unchanged under replacing f (x) by πf (x). Thus the distinction here is irrelevant. However, for a cluster s, the functions ν s and ǫ s (see (13.7), Definition 13.11) are defined with reference to the leading coefficient of the polynomial in question, hence may change upon replacing f (x) by πf (x). For example, for a proper cluster s, ν s is one larger when s is viewed as a cluster for C L than when it is viewed as a cluster for C.
In several statements below we will need to distinguish the following special case.
Notation 14.2. We say that R is atypical if R = s 1 ⊔ s 2 is a disjoint union of 2 odd proper clusters s 1 and s 2 , with both δ s 1 and δ s 2 odd.
The description of the special fibre of the minimal regular model of C L that we present below follows from work of Faraggi-Nowell [FN20], which more generally gives an explicit description of the special fibre of the minimal regular SNC model for hyperelliptic curves with tame reduction (that is, attaining semistable reduction after a tamely ramified extension of the base field). As is apparent from the statement of Proposition 14.5 below, their description simplifies significantly for quadratic twists of semistable hyperelliptic curves.
Remark 14.3. For an alternative, but related, approach to constructing regular models of hyperelliptic curves over nonarchimedean local fields of odd residue characteristic, see the works of Srinivasan [Sri19] and Obus-Srinivasan [OS19].
In what follows we denote by X the minimal regular model of C L over O K , and denote by Xk its special fibre, base-changed tok.
Notation 14.4. In describing Xk we will use the following terminology; see [FN20, Definition 3.1] for more details. By a chain of n rational curves of multiplicity d, n ≥ 0, d ≥ 1, we mean a collection of irreducible components Γ 1 , ..., Γ n of Xk, each isomorphic to P 1 k , such that Γ i intersects Γ i+1 transversally for each i, and such that each Γ i has multiplicity d in Xk. We depict this situation below. By a crossed tail, we mean a chain of rational curves Γ 1 , ..., Γ n , along with 2 additional irreducible components, the 'crosses', both isomorphic to P 1 k and intersecting Γ n transversally. Again, this situation is depicted below. In all the crossed tails we consider, each Γ i has multiplicity 2, whilst the crosses have multiplicity 1.
Chain of n rational curves of multiplicity d

Crossed tail
Proposition 14.5. All irreducible components of Xk intersect transversally, and no three components intersect at a point. Moreover: • every principal cluster s for C contributes to Xk a single component Γ s of genus 0 and multiplicity 2, • components corresponding to principal clusters s ′ < s are linked by: -a chain of 1 2 δ s ′ rational curves of multiplicity 1 if s ′ is odd, -a chain of (2δ s ′ − 1) rational curves of multiplicity 2 if s ′ is even, • for s principal, each twin t < s contributes a crossed tail T t whose first component intersects Γ s , and which consists of 2δ t rational curves of multiplicity 2, with the crosses having multiplicity 1, • if R = s 1 ⊔s 2 and both s 1 and s 2 are odd, then Γ s 1 and Γ s 2 are linked by a chain of 1 2 (δ s 1 +δ s 2 ) rational curves of multiplicity 1, • if R = s 1 ⊔ s 2 and both s 1 and s 2 are even, then Γ s 1 and Γ s 2 are linked by a chain of 2δ s 1 + 2δ s 2 − 1 rational curves of multiplicity 2, • for a principal cluster s, each child of size 1, {r} < s say, contributes a single rational curve T r of multiplicity 1, intersecting Γ s .
Proof. This essentially follows from specialising [FN20, Theorems 7.12 and 7.18] to the case in hand, noting that the minimal regular SNC model coincides with the minimal regular model in this case (more precisely, the description in the statement of this proposition is the description of the minimal regular SNC model of C L obtained from [FN20, Theorems 7.12 and 7.18], and visibly has no exceptional curves in its special fibre). The only catch is that the statements of [FN20, Theorems 7.12 and 7.18] contain some minor errors and as such one does not quite recover the description of Xk given above. A corrected version of these results appears in the PhD thesis of Nowell [Now22, Theorems 9.23, 9.31 and 9.32], from which one obtains the claimed statement. When applying the results cited above, recall that we are imposing Assumption 13.4. The necessary invariants which form the required input for [FN20, Theorems 7.12 and 7.18] and [Now22, Theorems 9.23, 9.31 and 9.32] are described in Lemma 14.7 below.
Caution 14.6. In the following lemma only, we view clusters as being associated to πf (x) rather than f (x), since it is invariants of the former which constitute the required input for the results of [FN20].
See [FN20, Table 3] and the references therein for the definitions of the invariants appearing in the following lemma. Briefly, for a proper cluster s for C L , the quantities d s , ν s and δ s are as defined in Section 13.1, but with πf (x) in place of f (x). By definition we have λ s = 1 2 ν s − d s s ′ <s ⌊ |s ′ | 2 ⌋ (we caution that this is the function denotedλ s in [DDMM18, Notation 1.19]). The quantity e s is the minimal integer such that both e s d s ∈ Z and e s ν s ∈ 2Z. When s is even, the invariant ǫ s ∈ {±1} in the statement is given by evaluating the function ǫ s of Definition 13.11 (which no longer factors through G k in general since C L /K is not semistable) at any topological generator of the tame inertia group of K. For our purposes we may take as a definition that ǫ s = (−1) ν s * −|s * |d s * for s * as in Notation 13.10. We will not use this variant of ǫ s anywhere else in the paper. Recall also from Notation 13.8 the definition of the set T .
Lemma 14.7. Let s be a proper cluster for C L (i.e. for the polynomial πf (x)). Then s is fixed by the inertia group I K of K, and all of the following hold: (i) we have d s ∈ Z unless s ∈ T , in which case d s ∈ 1/2 + Z, (ii) ν s is odd unless either s = R and R is atypical, or s ∈ T . In these cases, ν s is even.
(iii) if s is even then ǫ s = −1 unless s = R is atypical, in which case ǫ s = 1, (iv) if |s| ≥ 3 then e s = 2 unless s = R is atypical, in which case e s = 1, (v) if s is principal then λ s ∈ 1 2 + Z, (vi) if s ′ < s are principal clusters with s ′ odd, then δ s ′ is even.
Proof. As noted above, the proper clusters for C L and their associated depths are the same as those for C (and whether or not a cluster s is proper/principal/odd/even/übereven is similarly independent of whether we view s as a cluster for C or C L ). However, given a cluster s for C, when we view it as a cluster for C L the quantity ν s increases by 1 since the leading coefficient of πf (x) has valuation one greater than that of f (x). All claims are now a formal consequence of Theorem 13.6, which applies since C : y 2 = f (x) is semistable. Explicitly, the claim that each I K -orbit of proper clusters has size 1 is part (2) of Theorem 13.6. Part (i) is Lemma 13.9. Part (ii) for s / ∈ T is [DDMM18, Lemma 4.7], whilst for s ∈ T this follows from [DDMM18, Lemma C.5], combined with [DDMM18,Lemma 4.7] applied to the parent of s. Parts (iii) and (iv) follow from parts (i) and (ii). Part (v) follows from (ii). Finally, for part (vi) see [DDMM18,Lemma C.7].
It is convenient to package the description of Xk given in Proposition 14.5 in terms of the following graph. Remark 14.9. We see from Proposition 14.5 that T is a connected tree. Note that any vertex of T of degree at least 3 has weight 2. The leaves of T correspond to the components Γ r for r ∈ R not in a twin, along with the 'crosses' on the crossed tails T t for twins t. In particular, T has |R| = 2g + 2 leaves. Moreover, each leaf has weight 1.
Example 14.10 (Ramified quadratic twist of good reduction). Suppose f (x) ∈ O K [x] is monic, has degree 2g + 2 for some g ≥ 2, and is such that the reduction f (x) (mod π) is separable. Then C : y 2 = f (x) has good reduction, and C L /K is the hyperelliptic curve C L : y 2 = πf (x). We now use Proposition 14.5 to describe Xk in this case. The assumptions mean that f (x) has a single proper cluster, given by the full set of roots R. This cluster has depth 0 and has 2g + 2 children, with each individual root r ∈ R contributing a child {r} < R of size 1. The cluster picture is thus the following: 0 By Proposition 14.5, Xk consists of one component Γ R of genus 0 and multiplicity 2, intersected transversely by 2g + 2 rational curves of multiplicity 1, one for each root r ∈ R, as depicted below. The graph T consists of 2g + 2 vertices of weight 1, each joined to a common vertex v R of weight 2, as shown below also. In the picture we do not label multiplicities/weights unless they are greater than 1. Example 14.11. Take C to be the semistable genus 2 hyperelliptic curve over Q 3 considered previously in Examples 9.12 and 13.3, and take L = Q 3 ( √ 3), so that C L is the curve for i a square root of −1. As in Example 13.3 the cluster picture is as shown: As explained previously in Example 13.3, the full set of roots R is the unique principal cluster, and (as shown in the picture) there are 3 twins t 1 , t 2 and t 3 , with δ t 1 = 1/2 and δ t 2 = δ t 3 = 1. By Proposition 14.5, Xk consists of one component Γ R of multiplicity 2, along with 3 crossed tails, as depicted below. The corresponding graph T is pictured also.
Returning to the general case, we now describe the G k -action on the set of irreducible components of Xk (equivalently the induced G k -action on T ). To do this we introduce the following notation.
Notation 14.12. For each twin t = {r 1 , r 2 } let η t ∈ K s× be a choice of square root of (r 1 − r 2 ) 2 (−π) 2dt , noting that d t = v(r 1 − r 2 ) so that the displayed quantity is a unit (we can have d t ∈ 1 2 + Z, so we need not have a canonical choice of square root). In particular, η t ∈ O × K nr . Define the function γ t,L : G K → {±1} by the formula The function γ t,L factors through Gal(K nr /K). Thus we view γ t,L as a function on G k also. In particular, we can speak about γ t,L (F ) where F ∈ G k is the Frobenius element. The function γ t,L may depend on the choice of square root η t , but its restriction to the stabiliser of t does not. We remark that we include L in the notation for γ t,L since, when d t ∈ 1/2 + Z, it depends on the class of the uniformiser π in K × /K ×2 .
We stress that Convention 14.1 is in place, which is relevant for the function ǫ s .
Proposition 14.13. Let σ ∈ G k . The action of σ on the set of irreducible components of Xk is determined by: • for s principal, the component Γ s is sent to Γ σs , • for each r ∈ R not in a twin, the component Γ r is sent to Γ σr , • for a twin t with t / ∈ T , the crossed tail T t is sent to γ t (σ)T σt , 7 • for a twin t ∈ T , the crossed tail T t is sent to ǫ t (σ)γ t (σ)T σt .
Example 14.15. Returning to Example 14.11, for all σ ∈ G k we have ǫ t 1 (σ) = ǫ t 2 (σ) = ǫ t 3 (σ) = 1 (cf. Example 13.14). One checks that we may take the functions γ t 1 , γ t 2 and γ t 3 to be identically 1 also. Finally, the Frobenius element in G k fixes t 1 but swaps t 2 and t 3 . We thus see from Proposition 14.13 that F fixes the crossed tail corresponding to t 1 (the leftmost one in the picture) and swaps the crossed tails corresponding to t 2 and t 3 . Moreover, F 2 acts trivially on the full set of components, hence the stabiliser in G k of t i , i = 2, 3, acts trivially on the crosses of the corresponding crossed tail.
14.2. The Tamagawa number up to squares. We now use the description of T along with its G k -action, afforded by Propositions 14.5 and 14.13, to compute the Tamagawa number of J L /K up to rational squares. We begin by describing the order of the component group overk.
Proof. Since T is a tree, [BLR90, Proposition 9.6.6] gives where deg(v) denotes the degree of the vertex v. Since T is a connected tree we have Since any vertex of T of degree at least 3 has multiplicity 2, we find v∈T dv=1 the second equality following from Remark 14.9.
We now turn to computing the size of the G k -invariants of Φ(k) up to rational squares, which we will do with the aid of Proposition 10.8. We remark that an alternative appraoch might be to use the recipe [Sri16, Section 4.2] of Srinivasan for computing the Tamagawa number of a curve in terms of its minimal proper regular model.
In what follows it will be convenient to work exclusively with the graph T . To facilitate this, we transfer the intersection pairing between the components of Xk to a pairing on the vertices of T . Since by Proposition 14.5 all components intersect transversally, this pairing has a simple combinatorial description.
Note that if vertices v, v ′ of T correspond to components Γ v and Γ v ′ respectively, then v • v ′ is the intersection number between Γ v and Γ v ′ . We extend this product bilinearly to the Q-vector space V with basis the vertices of T .
Notation 14.18. For v ∈ T denote by r v the size of the G k -orbit of v. If r v is even write We now define a matrix M with rows and columns indexed by the even length G k -orbits of vertices of T as follows. For each even length G k -orbit O, pick a representative v O ∈ O. Then the (O, O ′ )-entry of M is defined as The relevance of the above definitions is that, by Proposition 10.8, we have Proposition 14.20. Suppose that either R is a principal cluster, or R = s 1 ⊔ s 2 is a disjoint union of two proper clusters s 1 and s 2 which are not swapped by G k . Then | det M | = 2 #{even sized G k -orbits of leaves of T } .
We begin with a lemma, which is a variant of [BLR90, Lemma 9.6.7].
Lemma 14.21. Let T be a rooted tree with root R. Let N be a matrix with rational coefficients whose rows and columns are indexed by the vertices of T. Suppose that N v,v ′ = 0 unless either v = v ′ or v and v ′ are adjacent in T. Further, suppose that all rows of N sum to 0, save possibly the row corresponding to the root R. Then where here for a vertex v = R of T, v parent denotes the parent of v in T (that is, the vertex adjacent to v on the unique path in T from v to the root R).
Proof. The strategy of proof is the same as that of [BLR90, Lemma 9.6.7], and is by induction on n = |T|. If n = 1 the result is clear, so assume n > 1. Let v = R be a leaf of T and order the vertices of T so that v is the first vertex, and its parent v ′ is the second (the determinant is independent of the ordering of vertices, this is just to enable us to write down the matrix explicitly). The assumptions on N mean that it has the form If N v,v ′ = 0 then detN is as claimed, so suppose N v,v ′ = 0. Adding column 1 to column 2 and then N v,v ′ · (row 1) to row 2 does not change the determinant, and transforms the matrix above into the matrix  Here all entries indicated by a * remain unchanged from the corresponding entries of N . Let N be the matrix obtained by removing the first row and column from this matrix, so that detN = −N v,v ′ det N .
Letting T be the rooted tree obtained from T by removing the leaf v (with root equal to the root R of T) we see that N satisfies the hypothesis of the statement with respect to T. By induction as desired.
Proof of Proposition 14.20. If R is principal, denote by R the vertex of T corresponding to the component Γ R . If R = s 1 ⊔ s 2 denote by R the vertex of T corresponding to Γ s 1 . In either case, R is fixed by G k and d R = 2. We view T as a rooted tree with root R. For a vertex v = R we denote by P (v) the parent of v in T . We say that v is a child of a vertex w if w = P (v). Now take a vertex v of T with r v even, noting that this forces v = R. If v ′ is a child of v then r v divides r v ′ . In particular, r v ′ is even also. In this case we write To make use of these computations, pick compatibly a representative for each even sized G k -orbit of vertices in T in such a way that if v is picked, then for each G k -orbit containing a child of v, the chosen representative of that orbit is itself a child of v. The subgraph of T generated by all chosen representatives is a finite disjoint union of connected trees, T 1 , ..., T s say. Each T i is naturally a rooted tree, with root the unique vertex of T i closest to R, and we extend the notion of child/parent to T i . We caution however that we reserve the notation P (v) for the parent of a vertex v in the tree T . Now for 1 ≤ i ≤ s, define N i to be the matrix whose rows and columns are indexed by the vertices of T i , and such that the (v, v ′ )-entry of N i is given by otherwise, the second equality following from (14.22) and (14.23). By construction we have Applying Lemma 14.21 to each of the matrices N i we find Claim: For each 1 ≤ i ≤ s we have Proof of claim. First suppose that T i consists of the single vertex R i . Then R i is necessarily a leaf in T , hence has weight 1 and parent (in T ) of weight 2. Thus the formula holds in this case. Now assume that T i consists of at least 2 vertices, and for a vertex v in T i let deg T i (v) denote the degree of v when viewed as a vertex of T i (as opposed to a vertex of T ). Note that a vertex Since T i is a connected tree we have On the other hand, if v ∈ T i has deg T i (v) ≥ 3 then v necessarily has degree at least 3 when viewed as a vertex of T . It then follows from the description of T afforded by Proposition 14.5 that d v = 2. All together, this gives Under the assumption that T i has at least 2 vertices, we see that a vertex v ∈ T i is a leaf in T if and only if deg T i (v) = 1 and v = R i . When this is the case, v necessarily has weight 1. This observation combined with (14.26) proves the claim (note that if deg T i (R i ) > 1 then, since R i = R, we see that R i must have degree at least 3 in T , hence weight 2).
Returning to the proof of the proposition, note that the number of leaves of T which appear in some T i is precisely the number of even sized G k -orbits of leaves of T . Combining the claim with (14.25) thus gives For each 1 ≤ i ≤ s the parent of R i lies in an odd sized G k -orbit and has a child lying in an even sized G k -orbit. In particular, either R i = R or R i has degree at least 3 in T . Either way, d R i = 2 and the result follows.
In the remaining case, when R = s 1 ⊔ s 2 is a disjoint union of 2 principal clusters swapped by G k , the result is the following.
Proposition 14.28. Suppose that R = s 1 ⊔ s 2 is a disjoint union of 2 principal clusters s 1 and s 2 that are swapped by G k . Then | det M | = 2 #{even sized G k -orbits of leaves of T } g odd, or g even and R not atypical, 1 2 · 2 #{even sized G k -orbits of leaves of T } g even and R atypical.
Proof. We indicate how to adapt the proof of Proposition 14.20 to these cases. First suppose that g is odd. Then both s 1 and s 2 are even. The vertices of T corresponding to Γ s 1 and Γ s 2 are joined by a path consisting of an odd number of vertices of multiplicity 2. Let R be the middle vertex in this path, which is fixed by G k and has multiplicity 2. With this definition of R, the proof of Proposition 14.20 applies verbatim to give the desired result. Now suppose g is even, so that s 1 and s 2 are both odd. Since s 1 and s 2 are swapped by G k we have δ s 1 = δ s 2 , and the vertices of T corresponding to Γ s 1 and Γ s 2 are joined by a path consisting of δ s 1 vertices of weight 1. If R is atypical then this path consists of an odd number of vertices, and we take as a root R the middle vertex in this path. This is fixed by G k and has weight 1. Following the proof of Proposition 14.20, (14.29) becomes (14.29) | det M | = 2 #{even sized G k -orbits of leaves of T } · d R 2 .
To see this, note that every vertex of T other than R has an even sized G k -orbit, so that s = 1 in the proof of Proposition 14.20. The result now follows immediately. Finally, suppose that g is even but that R is not atypical, so that the vertices of T corresponding to Γ s 1 and Γ s 2 are joined by a path consisting of a (positive since δ s 1 ≥ 1) even number of vertices. Let R 1 and R 2 be the middle vertices on this path, noting that they have degree 2 in T and weight 1. Note that every vertex of T has an even sized G k -orbit. As in the proof of Proposition 14.20, we compatibly pick a representative for each (even sized) G k -orbit of vertices in T , starting with R 1 , and in such a way that if v is picked then, for each G k -orbit containing a child of v, the chosen representative of the orbit is itself a child of v. Let T 1 be the subtree of T generated by the chosen vertices. This is a connected tree and we take R 1 as a root for T 1 . As in the proof of Proposition 14.20, define N 1 to be the matrix whose rows and columns are indexed by the vertices of T 1 and such that the (v, v ′ )-entry of N 1 is given by (N 1 ) v . This time, the formula (14.24) is valid provided (v, v ′ ) = (R 1 , R 1 ). Noting that ǫ R 1 = R 1 − R 2 , one computes Again, the matrix N 1 satisfies the conditions of Lemma 14.21, and the row corresponding to R 1 sums to 2d 2 R 1 . Thus Now T 1 consists of at least 2 vertices, R 1 has degree 1 in T 1 and weight 1, and leaves of T 1 other than R 1 correspond bijectively to (necessarily even sized) G k -orbits of leaves in T , all of which have weight 1. Arguing as in the claim in the proof of Lemma 14.21 now gives the result.
Recall from Notation 13.24 that we set κ(C) = 1 if R = s 1 ⊔ s 2 is a disjoint union of two odd G k -conjugate clusters with both δ s 1 and δ s 2 odd, and set κ(C) = 0 otherwise. Putting everything together we obtain the following.
As in Remark 14.9 there are 2g + 2 leaves of T . By Remark 14.14, the leaves corresponding to elements r ∈ R, together with leaves arising as the crosses on the crossed tails T t for twins t / ∈ T , form a G k -set isomorphic to R ∩ K nr . These leaves give rise to the second term on the right hand side of the statement. The remaining leaves arise as the crosses on the crossed tails T t corresponding to twins t ∈ T . Using Proposition 14.13 once again we see that each G k -orbit O ⊆ T gives rise to a single even sized G k -orbit of leaves if t∈O ǫ t (F )γ t,L (F ) = −1, and either 0 or 2 such orbits if t∈O ǫ t (F )γ t,L (F ) = 1 (according to whether |O| is odd or even). This gives the result.
15. Proof of Proposition 11.1 For this section we take K to be a non-archimedean local field of odd residue characteristic, take L = K( √ π) to be a ramified quadratic extension of K, and let C/K be a semistable hyperelliptic curve. We now combine the results of Sections 11-14 to prove Proposition 11.1. For convenience, we recall the statement.
Proof. By Lemma 4.1, we are at liberty to replace K with an arbitrarily large odd-degree unramified extension. In particular, we can without loss of generality assume that C/K is given by an equation of the form y 2 = f (x) where f (x) satisfies Assumption 13.4 (see Remark 13.5 for a justification of this). This allows us to use the results of Section 13.4 and Section 14 which were proven under this simplifying assumption. Combining Corollary 13.25 and Corollary 14.30 with (11.2) and (11.8) gives w(J/L) · (−1) ǫ(C/K)+ǫ(C L /K)+dim J(K)/N L/K J(L) = (−1) # even-sized G k -orbits on R∩K nr ·(−1) # G k -orbits O⊆T with t∈O γ t,L (F )=−1 .
Here we are using the notation of Sections 13 and 14, so that R denotes the set of roots of f (x) in K s , the set T is as defined in Notation 13.8, and the signs γ t,L are as defined in Notation 14.12. To prove Conjecture 1.7 we see that it suffices to establish the equality (15.2) (∆ C , L/K) ? = (−1) # even-sized G k -orbits on R∩K nr +# G k -orbits O⊆T with t∈O γ t,L (F )=−1 .
Recall from Lemma 13.9 that we have ∪ t∈T t = R R ∩ K nr , and that each t = {r t,1 , r t,2 } is an intertia-orbit of roots of f (x). In particular, we can factor f (x) over K as a product where f nr (x) ∈ K[x] splits over K nr and where, for a G k -orbit O ⊆ T , we have In what follows, for a polynomial g(x) we write ∆ g for its discriminant. From the above factorisation we find This follows from the fact that, for coprime polynomials h 1 (x), h 2 (x) ∈ K[x], we have ∆ h 1 h 2 = ∆ h 1 ∆ h 2 Res(h 1 , h 2 ) 2 where Res(h 1 , h 2 ) ∈ K × denotes the resultant of h 1 (x) and h 2 (x).
Since L/K is ramified whilst f nr (x) splits over an unramified extension, we see that (∆ f 0 , L/K) = 1 if and only if ∆ fnr is a square in K, which in turn happens if and only if the Frobenius element F ∈ G k acts as an even permutation of the roots of f nr (x). Thus we have (∆ fnr , L/K) = (−1) # even-sized G k -orbits on R∩K nr . To conclude, we claim that for each G k -orbit O ⊆ T we have (∆ f O , L/K) = t∈O γ t,L (F ). Indeed, from the definition of γ t,L given in Notation 14.12, we see that t∈O γ t,L (F ) is equal to 1 if and only if the quantity t∈O (r t,1 − r t,2 ) 2 (−π) −2dt ∈ O × K is a square in K. We thus have t∈O γ t,L (F ) = t∈O (r t,1 − r t,2 ) 2 (−π) −2dt , L/K = t∈O (r t,1 − r t,2 ) 2 , L/K , where for the second equality we note that −π is a norm from L = K( √ π) (recall that, since t ∈ T , the quantity 2d t is an odd integer). For t = t ′ ∈ O write R(t, t ′ ) = r∈t,r ′ ∈t ′ (r − r ′ ), noting that this quantity lies in K nr and that R(t, t ′ ) = R(t ′ , t). Then we have where the second product runs over all unordered pairs of distinct elements of O. The product {t,t ′ }⊆O R(t, t ′ ) is visibly fixed by G K , hence lies in K. We conclude that ∆ f O and t∈O (r t,1 − r t,2 ) 2 are congruent modulo squares in K × , proving the claim.

Residue characteristic 2
In this section we consider Conjecture 1.7 when K is a finite extension of Q 2 and when the quadratic extension L/K is ramified. Let C/K be a hyperelliptic curve with Jacobian J. We suppose henceforth that J/K has good ordinary reduction. Let J(K) 1 denote the kernel of reduction on J(K), and define J(L) 1 similarly. We begin by considering the norm map from J(L) 1 to J(K) 1 .
Proof. Let G = Gal(L/K) ∼ = Z/2Z. Let g be the genus of C so that by [LR78, Theorem 1], there is a matrix u ∈ Mat g (Z 2 ) (the twist matrix associated to the formal group of J) such that J(K) 1 /N L/K J(L) 1 ∼ = G g /(1 − u)G g . Moreover, denoting by T the completion of K nr we have (see [LR78,Lemma]) where F denotes the Frobenius automorphism of T . In particular, we have J(K) 1 [2] ∼ = α ∈ {±1} g : (1 − u)α = 1 . Identifying the groups G and {±1} in the obvious way, J(K) 1 [2] is identified with the kernel of multiplication by 1 − u on G g . We now conclude by noting that the cokernel and kernel of an endomorphism of a finite abelian group have the same order.  For the purpose of giving examples we now describe how to construct hyperelliptic curves over Q whose Jacobians are good ordinary over Q 2 and have all their 2-torsion defined over an odd degree extension on Q 2 . Let g ≥ 2 be an integer.
Lemma 16.4. Let f (x) ∈F 2 [x] be a monic separable polynomial of degree g + 1 and let h(x) ∈F 2 [x] be a polynomial of degree ≤ g, coprime to f (x). Then the hyperelliptic curve is ordinary.
Proof. One readily checks that the equation defining C is smooth, hence defines a hyperelliptic curve overF 2 of genus g. Let J be the Jacobian of C. As in the proof of [CST14, Theorem 23] one sees that dim J(F 2 )[2] = g, hence J is ordinary.
Lemma 16.5. Suppose f (x) ∈ Z[x] has odd leading coefficient and degree g + 1, and suppose that f (x) (mod 2) is separable with each irreducible factor having odd degree. Further, let h(x) ∈ Z[x] have degree ≤ g be such that h(x) (mod 2) is coprime to f (x) (mod 2). Then the Jacobian J of the hyperelliptic curve C : y 2 = f (x)(f (x) + 4h(x)) has good ordinary reduction over Q 2 . Moreover, Q 2 (J[2])/Q 2 has odd degree.