Parity conjecture for abelian surfaces

Assuming finiteness of the Tate--Shafarevich group, we prove that the Birch--Swinnerton-Dyer conjecture correctly predicts the parity of the rank of semistable principally polarised abelian surfaces. If the surface in question is the Jacobian of a curve, we require that the curve has good ordinary reduction at 2-adic places.


Main results
The Birch-Swinnerton-Dyer conjecture predicts that the Mordell-Weil rank of an abelian variety over a number field is given by the order of vanishing of the -function ( ∕ , ) at = 1. Despite being more than half a century old, there has been little theoretical evidence for the conjecture beyond the case of elliptic curves. The aim of the present article is to show that it correctly predicts the parity of the rank of abelian surfaces, at least if one is willing to assume the finiteness of Tate-Shafarevich groups.
Recall that the functional equation for ( ∕ , ) says that this function is essentially either symmetric or antisymmetric around the central point =1, and, consequently, the sign in the functional equation determines the parity of the order of the zero there. Of course, neither the analytic continuation of the -function nor its functional equation have been proved. However, part of the conjectural framework specifies that the sign is given by the global root number ∕ ∈ {±1}, an invariant that is defined independently of any conjectures. One thus expects that the root number controls the parity of the rank of ∕ : Our main result is the following: Theorem 1.2 (=Theorem 13.5). The parity conjecture holds for principally polarised abelian surfaces over number fields ∕ such that X ∕ ( [2]) has finite 2-, 3-and 5-primary part that are • Jacobians of semistable genus 2 curves with good ordinary reduction at primes above 2, or • semistable, and not isomorphic to the Jacobian of a genus 2 curve.
We note that the hypothesis at primes above 2 requires the underlying curve, and not merely the Jacobian itself, to have good reduction. By a curve with 'ordinary' reduction, we mean one whose Jacobian has ordinary reduction.
There is a range of results on the parity conjecture in the context of elliptic curves, but the progress for higher dimensional abelian varieties has been rather limited. Previous results only apply to sparse families, for example, [8] requires the abelian variety to admit a suitable -rational isogeny of degree dim , and [26] addresses Jacobians of hyperelliptic curves that have been basechanged from a subfield of index 2.
The proof of the above theorem has two main ingredients. The first is the following reduction step that applies to abelian varieties of arbitrary dimension. Its proof is based on the method of regulator constants of [11,12] and will be explained in Appendix B. Theorem 1.3. Let ∕ be a Galois extension of number fields with Galois group . Let ∕ be a semistable principally polarised abelian variety such that X ∕ is finite. If the parity conjecture holds for ∕ for all 2-groups ⩽ , then it holds for ∕ .
The second ingredient is a proof of Theorem 1.2 under the assumption that the degree of the field extension generated by [2] is a power of 2. More precisely, we establish the '2-parity conjecture' in this case. Without some assumption on the Tate-Shafarevich group, the parity conjecture currently appears to be completely out of reach -indeed, it would give an elementary criterion for predicting the existence of points of infinite order, something that seems to be impossibly difficult already for elliptic curves. However, the version for Selmer groups is more tractable. We will write rk ( ∕ ) for the ∞ -Selmer rank of , that is, rk ( ∕ ) = rk( ∕ ) + , where is the multiplicity of ℚ ∕ℤ in the decomposition X ∕ [ ∞ ] ≃ (ℚ ∕ℤ ) ×(finite) and is conjecturally always 0. • Jacobians of semistable genus 2 curves with good ordinary reduction at primes above 2, or • not isomorphic to the Jacobian of a genus 2 curve.
Assuming the finiteness of Tate-Shafarevich groups, the -parity conjecture clearly implies the parity conjecture. In particular, Theorem 1.2 is a direct consequence of Theorems 1.5 and 1.3 applied to = ( [2]).
The proof of Theorem 1.5 consists of two parts, outlined in more detail in §1.2 and §1.3 below. The first expresses the parity of the 2 ∞ -Selmer rank of ∕ as a product of some local terms ∕ , analogously to the formula for the global root number as a product of local root numbers ∕ = ∏ ∕ , the products taken over all the places of . This makes crucial use of a Richelot isogeny on whose existence is guaranteed by the restriction on Gal( ( [2])∕ ).
The second part is the proof that this expression for the parity of the rank is compatible with root numbers. In other words, that ∕ ∕ satisfies the product formula which leads to the desired expression (−1) rk 2 ∕ = ∕ . This product formula is more delicate than one might expect, because one often has ∕ ≠ ∕ . However, rather miraculously, ∕ ∈ {±1} always differs from ∕ ∈ {±1} at an even number of places . Conjecture 1.14 below gives an explanation for this phenomenon, by describing an explicit relation between the two local invariants. The key point of the conjecture is that it reduces the global problem of controlling the parity of 2-Selmer ranks to the purely local one of proving an identity between various invariants of genus 2 curves defined over local fields. We prove this conjecture for all semistable curves with good ordinary reduction at primes above 2 (see Theorem 1.16), which let us deduce the 2-parity result of Theorem 1.5 and hence Theorem 1.2 (see Theorem 1.15). The proof relies on explicit formulas and the study of genus 2 curves over local fields, and occupies a substantial part of the present paper.
If is not centred and ′ is the centred curve corresponding to it by shifting the -coordinate, we define these quantities for as being those for ′ , that is, Δ( ) = Δ( ′ ), and so on.
Manipulating formal algebraic expressions such aŝ∕ with a computer is not practical: these expressions are enormous and computers cannot simplify Hilbert symbols. However, we made extensive use of computational data to find the expression for ∕ . Once one finds the right list of invariants , it is not difficult to produce the product expression of ∕ : one compiles a large list of C2D4 curves and for each curve, one computes Jac ∕ , ∕ and all possible Hilbert symbols ( , ). One then uses linear algebra to find an expression for Jac ∕ ∕ in terms of these Hilbert symbols. The difficulty is then to find this list of invariants in the first place, the main issue being that Hilbert symbols do not behave sensibly under addition. Classical invariants such as Igusa invariants are not sensitive to Richelot isogenies and some of the local data that determine Jac ∕ ∕ . Our invariants carry this information, for example, see proof of Theorem 5.2 and §9. 1. In principle, Jac ∕ ∕ only depends on the Richelot isogeny; in terms of Definition 1.11, it means that it is symmetric in , and . However, there appears to be a barrier to finding a Hilbert symbol expression for Jac ∕ ∕ without breaking this symmetry or the symmetry between and̂.
We have numerically verified Conjecture 1.14 on all 40441 genus 2 curves currently in the LMFDB whose simplified model is given by a degree 6 polynomial, for all odd primes of tame reduction and for each possible C2D4 structure for which , Δ ≠ 0 (excluding the small number of cases when Magma failed to return a regular model for̂). In theory, one might be able to prove this conjecture over a specific local field by numerically checking a finite list of curves in the vein of Halberstadt's work on root numbers ( [20]). However, the length of the list is likely to be unreasonable.

Overview
In §2, we review background material, including the construction of the Richelot dual curve and the theory of clusters of [15], which will allow us to control local invariants of genus 2 curves over completions for primes of odd residue characteristic. In §3, we explain how to control the parity of the 2 ∞ -Selmer rank for Jacobians of curves that admit a suitable isogeny, and prove a general version of Theorem 1.8 (see Theorem 3.2). We also prove a formula for ∕ , which converts the kernel-cokernel into Tamagawa numbers and other standard quantities (Theorem 3.2); for example, for curves over finite extension of ℚ with odd, it reads ∕ =ˆ(−1) ord 2 ∕ ∕ˆ∕ , where andˆare the two Jacobians.
Sections §4-12 focus on C2D4 curves and form the technical heart of the proof of Theorem 1.5 on the 2-parity conjecture and Theorem 1.16 on Conjecture 1.14, which compares local root numbers to the -terms. Roughly, the idea is the following.
First of all, we can work out certain cases by making all the terms in Conjecture 1.14 totally explicit. For example, suppose that ∕ℚ is a C2D4 curve for ≠ 2, given by 2 = ( ) with ( ) ∈ ℤ [ ] monic, and that ( ) mod has four simple roots̄2,̄2,̄3,̄3, and a double root 1 =̄1. The reduced curve has a node, and, analogously to multiplicative reduction on an elliptic curve, the Jacobian has local Tamagawa number ( 1 − 1 ) 2 = ( 1 ) if the node is split, and 1 or 2 (depending on whether ( 1 ) is odd or even) if the node is non-split. Whether the node is split or non-split turns out to be precisely measured by whether or not is a square in ℚ . An explicit computation of the Richelot dual curve shows that, generically (if (Δ) = 0), its reduction also has a node and its Jacobian's Tamagawa number is 2 ( 1 ) or 2 depending again on whether is a square (split node) or not (non-split node). Neither curve here is deficient, so we obtain ∕ = −1 unless is a non-square and ( 1 ) is even, in which case it is +1. As for multiplicative reduction on elliptic curves, the local root number in this case is Jac ∕ℚ = ±1 depending on whether the node is split (−1) or non-split (+1). Finally, generically (!) all the terms apart from 1 in the expression for ∕ℚ are units, so that all the Hilbert symbols are (unit,unit)= 1, except for one remaining term ( , 1 ). The latter is −1 precisely when is a non-square (non-split node!) and ( 1 ) is odd. This magically combines to = ⋅ , as required.
We will work out a number of cases by a similar brute force approach ( §8-10); this is often rather more delicate than described above, as we have brushed the non-generic cases (when certain quantities become non-units) under the rug. Unfortunately, there is a myriad of possible reduction types that one would need to address to prove the formula = ⋅ in general. Instead, we will use a global-to-local trick to cut down the number of cases to a manageable list (from 938 to 48, in the description used in Theorem 7.1). This is based on the following lemma, which follows directly from Theorem 1.8 and the product formula for Hilbert symbols. Lemma 1.19. Let be a number field and ∕ a C2D4 curve with , Δ ≠ 0 for which the 2-parity conjecture holds. If Conjecture 1.14 holds for ∕ for all places of except possibly one place , then it also holds for ∕ .
Thus, to prove the formula = ⋅ of Conjecture 1.14 for over a local field, we can try to deform it to a suitable curve over a number field. The main difficulty, of course, is that we do not a priori have a supply of C2D4 curves over number fields for which we know the 2-parity conjecture! However, we can bootstrap ourselves by making use of the cases for which we have worked out Conjecture 1.14 using the brute force approach outlined above, and which give us a supply of C2D4 curves over number fields for which the 2-parity conjecture holds. Observe also that the truth of the 2-parity conjecture for a C2D4 curve is • independent of the choice of model for and • independent of the choice of the C2D4 structure.
This will let us show that Conjecture 1.14 is also independent of the choice of model and the choice of C2D4 structure for curves over local fields.
To make this method work, we need to understand how various quantities behave under a change of model ( §6), and to have a way to approximate C2D4 curves over local fields by C2D4 curves over number fields that, moreover, behave well at all other places ( §11). In §12, we justify that these tools are enough to prove Conjecture 1.14 in all the cases we claim in Theorem 1. 16.
In §13, we tie these results together, deal with the exceptional cases when  = 0, Δ = 0 or the abelian surface is not a Jacobian and prove Theorems 1.2 and 1.5. Appendix A (by A. Morgan) provides a formula for for curves with good ordinary reduction over 2-adic fields. Appendix B (by T. Dokchitser and V. Dokchitser) deals with regulator constants and Theorem 1.3.

General notation
Throughout the paper, rk ( ∕ ) will denote the ∞ -Selmer rank of ∕ (see Conjecture 1.4) and the dual of a given isogeny . For a local field with residue field , a curve ∕ and an abelian variety ∕ , we write uniformiser of a local non-archimedean field We will write ( ), Δ( ), and so on, if we wish to stress which curve we are referring to.
Remark 2.2. When 1 , 2 or 3 = 0, one can define the Richelot dual curve by the same construction by cancelling the offending terms in the equation forˆand the expressions for̂,̂,̂. [27,Corollary 12]). A curve of genus g over a local field is deficient if it has no -rational divisor of degree g −1. For a genus 2 curve ∕ , being deficient is equivalent to not having any -rational points over all extensions ∕ of odd degree.

Pictorial representation of roots
Notation 2.4. For a C2D4 curve ∶ 2 = ( ) ( ) ( ), we pictorially represent the roots 1 , 1 of ( ) as ruby circles ( ), roots 2 , 2 of ( ) as sapphire hexagons ( ) and roots 3 , 3 of ( ) as turquoise diamonds ( ). We will sometimes refer to them as ruby, sapphire and turquoise roots, respectively. Note that the Galois group ⩽ 2 × 4 preserves the set of ruby roots and either preserves the set of sapphire roots and the set of turquoise roots, or swaps these two sets around. Notation 2.5. For a C2D4 curve ∕ℝ, it will turn out that most of the local data that we are interested in are encoded in the arrangement of the real roots of the defining polynomial on the real line. We will depict this information by drawing the real roots in the order that they appear in ℝ and connect two roots , ′ if the points ( , 0) and ( ′ , 0) are on the same connected component of (ℝ). Thus, for example, a curve with 1 < 1 < 2 < 2 < 3 < 3 and < 0 will be depicted by .

Clusters: Curves over local fields with odd residue characteristic
To keep track of the arithmetic of genus 2 curves over -adic fields with odd, we will use the machinery of 'clusters' of [15].
Definition 2.6 (Clusters). Let be a finite extension of ℚ and ∶ 2 = ( ) a genus 2 curve, where ( ) ∈ [ ] is monic of degree 6 with set of roots . A cluster is a non-empty subset ⊂  of the form = ∩  for some disc = { ∈̄| ( − )⩾ } for some ∈̄and ∈ ℚ. For a cluster of size > 1, its depth is the maximal for which is cut out by such a disc, that is, =min , ′ ∈ ( − ′ ). If moreover ≠ , then its relative depth is = − ( ) , where ( ) is the smallest cluster with ⊊ ( ) (the 'parent' cluster). We refer to this data as the cluster picture of . For C2D4 curves, we will often specify which roots are ruby, sapphire and turquoise: we will refer to this data as the colouring of the cluster picture. Notation 2.7. We draw cluster pictures by drawing roots ∈  as , or as in Notation 2.4 if we wish to specify which root is which, and draw ovals around roots to represent clusters (of size > 1), such as: The subscript on the largest cluster  is its depth; on the other clusters, it is their relative depth. Proof. [15,Cor. 15.3]. □ Lemma 2.11. Let be a finite extension of ℚ for an odd prime . If ∕ is a balanced centred C2D4 curve, then ( ), ( ) ⩾ 0 for = 1, 2, 3. Moreover, if ( ) = 0 for some , then (̂), (̂) ⩾ 0.
Proof. Since the curve is balanced and centred, 1 = − 1 and the depth of the top cluster is 0. Thus, ( 1 ) = ( 1 2 ( 1 − 1 )) ⩾ 0. The first claim follows as ( 1 − ) ⩾ 0 for each root . The second claim follows directly from Definition 2.1, aŝ,̂are roots of a monic quadratic polynomial with integral coefficients. □ Roughly speaking, the proof of the formula = of Conjecture 1.14 will require a separate computation for each balanced cluster picture.

2.6
Local invariants of semistable curves of genus 2 Let ∕ be a curve of genus 2 over a finite extension of ℚ for an odd prime . We record some results of [15] that will let us control the arithmetic invariants of ∕ in terms of its cluster picture. (1) The extension ()∕ has ramification degree at most 2.
(3) Every principal cluster has ∈ ℤ and ( )+| | + ∑ ∉ ( − )∈2ℤ, for any (equivalently every) root ∈ . Here a cluster is principal if | | ⩾ 3, does not properly contain a cluster of size 4, and is not a disjoint union of two clusters of size 3 or of sizes 5 and 1.
We will need to keep track of the analogue of the split/non-split dichotomy for elliptic curves with multiplicative reduction. This is done by keeping track of the Galois action on clusters and associating signs ± to certain clusters of even size, see [15,Definition 1.13]. We will only need their explicit expressions for balanced pictures: Definition 2.13 (Sign and ). Suppose that is semistable and balanced.
(1) If  is a union of three twins, the sign of  is + if ∈ ×2 and − if ∉ ×2 .

Notation 2.15.
If we wish to keep track of the signs of clusters in our pictures of Notation 2.7, these will be written as superscripts to the ovals. If we wish to keep track of the Frobenius action, then lines joining clusters (of size > 1) will indicate that Frobenius permutes them. We refer to cluster pictures with this extra data as cluster picture with Frobenius action.
We will mostly use this table for balanced curves, that is, the first column of the cluster pictures. Note that type I , , is the same as I , , . Similarly, I × I is the same as I × I , and U , , is unchanged by any permutation of the indices. Remark 2.18. We will use a little more information about the types I , and I × I . Suppose that has type I , , or I × I . Write + and − for the cyclic group on which Frobenius acts trivially and by multiplication by −1, respectively. By [15,Thm. 1.15 and Lemma 2.22], the Néron component group of Jac ∕ is Φ = × . Note also that if has type +,− , and Φ = + × − , for some even and , then necessarily = and = . Indeed, if + × − ≃ + ′ × − ′ , then = ′ and = ′ , since the groups have 2 = 2 ′ Frobenius-invariant elements, and 2 = 2 ′ elements on which Frobenius acts by −1.

PARITY OF ∞ -SELMER RANK OF JACOBIANS WITH A g -ISOGENY
In this section, we discuss how to control the parity of the 2 ∞ -Selmer ranks for Jacobians of curves that admit a suitable isogeny. Definition 3.1. Let and ′ be curves over a local field whose Jacobians admit an isogeny ∶ Jac → Jac ′ with = [2] (equivalently, an isogeny whose kernel is a maximal isotropic subspace of Jac [2] with respect to the Weil pairing). We write where ker | = Jac ( )[ ] and coker | = Jac ′ ( )∕ (Jac ( )). For a C2D4 curve ∕ , this is ∕ of Definition 1.7.

Parity theorem
Theorem 3.2. Let and ′ be curves over a number field whose Jacobians admit an isogeny ∶ Jac → Jac ′ with = [2]. Then the product taken over all places of .
Proof. Write = Jac and ′ = Jac ′ . As in the proof of Thm. 4.3 in [12] where X nd ∕ denotes X ∕ modulo its divisible part. By a result of Poonen and Stoll ([27, Thm. 8, Cor. 12]), the order of |X nd ∕ | is a square if and only if is deficient at an even number of places, and is twice a square otherwise (and similarly for ′ ). Hence, where =2 if ∕ is deficient and =1 otherwise; and similarly for ′ . By definition of , the 2-adic valuation of the term at is even if and only if ∕ , = 1. The result follows. □

Kernel/cokernel on local points
The kernels and cokernels of the vertical maps are finite, so, by the snake lemma, The map on the connected component of the identity is surjective (as ≃ ℝ), and the groups of connected components are both finite, so this simplifies to the expression claimed. The case of non-archimedean is similar, with ( ) • replaced by 1 ( ), the kernel of the reduction on the Néron model of , see, for example, Lemma 3.8 in [29].

Odd degree base change
Finally, we record a basic observation regarding the behaviour of Conjecture 1.14 in odd degree unramified extensions.
is also unchanged up to squares.
The result thus follows from Lemma 3.4. □

MAIN LOCAL THEOREM: BASE CASES
We now turn to the proof of Conjecture 1.14, which relates local root numbers to the local term ∕ . As outlined in §1.4, we begin by proving a number of cases through explicit computation, summarised in Theorem 4.1. The proof will occupy §4- §10. In §11-12, we will deduce the conjecture for the general class of C2D4 curves in Theorem 1.16 by deforming them to number fields and using a global-to-local trick. Recall from §2.4, 2.5 that we draw pictures to indicate the distribution of the roots of ( ) in ℝ or ℚ .  (4) A substitution ↦ , ↦ 3 scales the roots by without changing the cluster picture or the leading term . This does not change any of the Hilbert symbols in ∕ ( , … all have even degree) nor ∕ . Thus, we may assume that the depth of the maximal cluster is 0. We may also assume that the C2D4 curve is centred, as a shift in the -coordinate does not change any of the invariants. Theorem 7.1 exhausts all possible Frobenius actions on these cluster pictures (after possibly recolouring ↔ ). By the semistability criterion (Theorem 2.12), ∈ ℤ for every cluster of size ⩾ 3 and Proof. Here ∕ℂ = 1 as has genus 2, and clearly, ∕ =1 and ∕ℂ = (−1) 2 =1. □ For curves over ℝ, we shall, for the moment, only prove Conjecture 1.14 in a restricted number of cases. The direct proof below can be extended to all cases (cf. [23]), but we will obtain the remaining ones for free using our methods in §11-12 (see Theorem 12.5 Proof. Write as 2 = ( ) ( ) ( ) as in Definition 1.11. Note that in cases (3)-(9), the picture indicates that < 0. We find that the number of components of (ℝ) is 0 in case (2) (Definition 2.1). Explicitly, with similar expressions for (̂2 −̂2) 2 and (̂3 −̂3) 2 , obtained by permuting the indices 1-3. If 2 , 2 , 3 , 3 ∈ ℝ, the above discriminant is positive if and only if the roots of the two quadratics are not interlaced (they are 'interlaced' if • 2 <• 3 <• 2 <• 3 or vice versa). If either 2 =̄2 or 3 =̄3, the discriminant is always positive, being of the form̄2 for ∈ ℝ. An identical analysis applies to (̂2 −̂2) 2 and (̂3 −̂3) 2 .
Putting this information together and considering the sign of the leading term, we deduce thatĥ as three real components in all cases above, except for case (8), when it has two real components. This gives the values forˆ∕ ℝ andˆ∕ ℝ as above for . ] to the identity (take a path to any ∈ ℝ). If , ∈ ℝ and ( , 0) and ( , 0) lie on the same component of (ℝ), then moving ( , 0) to ( , 0) along (ℝ) gives a path from [( , 0), ( , 0)] to the identity. However, if , ∈ ℝ and ( , 0) and ( , 0) do not lie on the same component of (ℝ), then no such path exist: both points have to remain in (ℝ) on the path as the -coordinates will never have the same real part and hence will never be complex conjugate. This fully determines the order of (ℝ) The formula for ∕ now follows from Lemma 3.4. As has genus 2, the Jacobian is 2-dimensional and ∕ℝ = (−1) dim = 1. Conjecture 1.14 for all the cases in the table will thus follow once we justify the formula for ∕ℝ .
We finally turn to ∕ℝ . This will be done by a case-by-case analysis of Hilbert symbols. For convenience, we may assume that the curve is centred, that is, 1 = − 1 , as (by definition) this does not affect any of the Hilbert symbols defining ∕ℝ .

CHANGING THE MODEL BY MÖBIUS TRANSFORMATIONS
For the proof of our main results on Conjecture 1.14, it will often be useful to be able to change the model of a C2D4 curve. This does not change the classical arithmetic invariants, but it does affect the terms Δ, , … that enter ∕ and hence Conjecture 1.14. In this section, we discuss possible changes of model and their effect on these terms.
Remark 6.2 (See also [22, §2]). If a genus 2 curve over admits two hyperelliptic models ∶ 2 = ( ) and ′ ∶ 2 2 = 2 2 ( 2 ), then the -coordinates are always related by a Möbius map 2 = ( ) = + + for some , , , ∈ (because these are the only transformations on ℙ 1 that is the quotient of the curve by the hyperelliptic involution). If both equations have degree 6, the model ′ then agrees with up to scaling the -coordinate by a suitable constant, 2 = for some ∈ .

Rebalancing
Theorem 6.5. Let be a finite extension of ℚ for an odd prime , with residue field of size | | > 5, and ∕ a semistable C2D4 curve. There is ∈ GL 2 ( ) such that is balanced.
Proof. Theorem 2.10 and Remark 6.2. □ Theorem 6.6. Let be a finite extension of ℚ for an odd prime , with residue field of size | | ⩾ 23. Let ∕ be a centred balanced semistable C2D4 curve. Then there is a 0 ∈ such that for all ∈ with ( − 0 )>0, the cluster picture of with signs and Frobenius action on proper clusters, its colouring, ( ) and (Δ) are the same as that of and Proof. Since is centred and balanced, all the roots are necessarily integral (Lemma 2.11). One readily checks that .
In particular, so long as ≢ −1∕ 1 , −1∕ 2 , 1∕ 1 , −1∕ 1 in , one necessarily has ( 1 − 2 ) = ( ( 1 ) − ( 2 )). Thus, if ≢ −1∕ in for any root , then has the same cluster picture as , with the same colouring. The Galois action on proper clusters and the signs of clusters is the same by Lemma 6.7 below. Moreover, the same condition on ensures that the valuation of the leading term and of Δ∕ remain unchanged (cf. Definition 6.1, Lemma 6.8(i) below).
Recall that (for ) 3 = 2 + 2 . Now Observe that the numerator is the zero polynomial in ( ) if and only if 2 ≡ − 2 and 2 2 ≡ 2 1 in . This is equivalent to 2 ≡ ± 1 and 2 ≡ ∓ 1 , which would mean that there is a cluster of depth > 0 containing 2 and ± 1 and one containing 2 and ∓ 1 . This is not the case for the listed cluster pictures, except for , so the numerator is not the zero polynomial in ( ) for these. It follows that, so long as avoids the roots of the polynomial in and the residues of −1∕ 2 , −1∕ 2 , the expression ( 2 ) + ( 2 ) will have valuation 0 in̄. Repeating a similar argument for 2 shows that has ( 2 ) = ( 3 ) = 0 so long as avoids a specific list of residue classes of . For the exceptional cluster picture , the coefficients of the numerator all have valuation ⩾ , and one similarly checks that at least one has valuation exactly , so that a suitable choice of makes The arguments for 1 and 2 , 3 are similar. Recall that (for ), 1 = 2 + 2 − 3 − 3 . Writing = 2 + 2 − 3 − 3 , = 2 2 − 3 3 and = 2 2 3 − 2 3 3 + 2 2 3 − 3 2 3 , one checks that and that if the numerator reduces to the zero polynomial in ( ), then {̄2,̄2} = {̄3,̄3}. This is not the case in the listed cluster pictures, except for and , and picking that avoids the residue classes that make the numerator or denominator 0 in makes ( 1 ( )) = 0. For the two exceptional cluster pictures, each coefficient has valuation ⩾ min( , ) (respectively, ⩾ ), and one easily checks that either or must have precisely this valuation. Picking similarly gives ( 1 ( )) = min( , ) (respectively, = ). For 2 , one finds that Here the numerator reduces to zero in ( ) only if 2 1 ≡ 2 2 ≡ 2 2 in , equivalently only if 2 ≡ ± 1 and 2 ≡ ± 1 (and similarly for 3 ). This only happens for and of the listed cluster pictures, which make no claim for 2 , 3 . Thus, will have ( 2 ) = ( 3 ) = 0, so long as avoids the residue classes that make either the numerators or denominators of 2 ( ), 3 ( ) reduce to 0 in .
The total number of residue classes has to avoid is at most 6 (of the form −1∕ for a root , that account for all the denominators) plus 2 + 2 (for 2 , 3 ) plus 4 (for 1 ) plus 4 + 4 (for 2 , 3 ), that is, 22. □ Lemma 6.7. Let be a finite extension of ℚ for an odd prime and =( )∈GL 2 ( ). Suppose that and are semistable, balanced C2D4 curves over , and that ↦ ( ) induces a bijection between the sets of twins and preserves their relative depths. Then ↦ ( ) also commutes with the Galois action on twins and preserves the signs of clusters of even size (after possibly suitably choosing signs of ( ) for twins of ).
Proof. Since , , , ∈ , the Galois action on the roots for is the same as on the roots on , and so, the map respects the Galois action on twins. It remains to check that it respects signs.
Suppose that has exactly one twin, 1 . As the two curves are isomorphic, Theorem 2.17 tells us that the sign must be the same for 1 and ( 1 ) (see types I and 1 × I ).
Suppose that has two twins, 1 and 2 . If these are swapped by Frobenius, then choosing the signs of ( ) appropriately guarantees that the sign of 1 agrees with that of ( 1 ); by Theorem 2.17, the signs of 2 and ( 2 ) must then also agree (see types I ∼ and I×I ). If the twins are not swapped but have the same sign, then the result again follows by Theorem 2.17 (types I +,+ , , I −,− , , I + × I + , I − × I − ). If the twins are not swapped and have different signs and different relative depths, the result follows from the structure of the Néron component group by Remark 2.18, after possibly passing to a quadratic ramified extension (types I +,− , , I + × I − with ≠ ). If the twins are not swapped by Frobenius and have different signs (say, + for 1 and − for 2 ) and equal relative depths (say = ), we unfortunately need to use the explicit description of the minimal regular models and the reduction map to the special fibre (see [15,Thm. 8.5 and §5.6]): passing to a quadratic ramified extension if necessary so that is even, the special fibre of the minimal regular model of is and Frobenius fixes the components on the left 2 -gon and acts as a reflection on the right 2gon. The two Weierstrass points of that correspond to the roots in the twin 1 reduce to the component in the left 2 -gon that is furthest away from the central chain, corresponding to the fact that the sign of 1 is +, and the two Weierstrass points for 2 reduce to the corresponding component in the right 2 -gon. As this description is model-independent, it follows that the Möbius transformation must preserve signs of the twins.
Finally, when and have three twins each, the only signs are those of the full sets of roots. These agree by Theorem 2.17 (types U * ). □

Change of invariants
Lemma 6.8. Let be a C2D4 curve over a field of characteristic 0.
Proof. Direct computation. □ Lemma 6.9. Let ∕ℚ be a finite extension with an odd prime, and let ∕ be a C2D4 curve with cluster picture 0 . Then for any ∈ GL 2 ( ) such that ∕ is balanced.
Proof. Enlarging if necessary, we may pick ∈ which has ( − ) = for the roots inside the cluster of size 5 of and ( − ) = 0 for the remaining root. One checks that applying the following Möbius transformation yields a model with a balanced cluster picture: ∶ ↦ − ↦ 1 − ↦ − . By Lemma 6.8, (Δ( )) − ( ( )) = (Δ) − ( ) − 5 + 3 . It remains to show that if and are both balanced models, then (Δ( )) = (Δ( )). As Δ is invariant under shifts of the -coordinate, we may assume that both and are centred; in particular, 1 ( ) and 1 ( ) are both units. By Lemma 6.4, the associate Möbius transformation is of the form = • for some , where ( ) = and = 1 ( ) 1 ( ) ∈ . As the roots are integral with distinct images in the residue field for both curves, we find that ≢ ± −1 1 , −1 2 , − −1 2 , −1 3 , − −1 3 in the residue field. The result now follows from Lemma 6.8(i). □ Lemma 6.10. For a C2D4 curve over a field of characteristic 0 and ∈ GL 2 ( ), Proof. Aŝ1 is invariant under shifts of the -coordinate, we may assume that both and are centred. By Lemma 6.4, the associate Möbius transformation is of the form = • for some ∈ ∪ {∞}, where ( ) = and = 1 ( ) 1 ( ) ∈ . As 2̂1 is a homogeneous rational function of even degree in the roots, with the natural extension of the formula to =∞. □

ODD PLACES
Here, we state an analogue of Theorem 5.2 for C2D4 curves over finite extensions of ℚ for odd primes . Its proof will occupy §8 and §9. We will remove the constraints on valuations in Theorem 12.2 and extend it to all semistable curves in Theorem 12.5. If the residue field has size | | ⩾ 23, thenˆadmits a model whose cluster picture with Frobenius action is given in theˆcolumn. ∕ is as given in the table provided , 1 ≠ 0 and: In the table , , , , ∈ ℤ >0 are parameters and ∈ ℤ is defined by the column of (Δ∕ ). In theĉ olumn, a cluster of size 3 with index 0 means that the roots in it do not form a cluster, for example, Remark 7.2. In the cases I , , ( ), the semistability criterion (Theorem 2.12) and the C2D4 structure on ensure that ≡ mod 2. Indeed, is odd if and only if inertia permutes the roots in the corresponding twin. The C2D4 structure then forces inertia to permute the roots in the twin of depth ∕2.

ODD PLACES: CHANGE OF INVARIANTS UNDER ISOGENY
In this section, we prove the claim of Theorem 7.1 regarding Tamagawa numbers and deficiency and the cluster picture ofˆwhen has odd residue characteristic. (2) If the residue field has size | | ⩾ 23, thenˆadmits a model whose cluster picture with Frobenius action is given in theˆcolumn of the table. Proof.
(2) ⇒ (1). The formulae for ∕ , ∕ and ∕ follow directly from Theorem 2.17. To determinê ∕ andˆ∕ , we may first pass to an unramified extension of sufficiently large degree so that | | ⩾ 23 (Lemma 3.7). As these invariants are independent of the choice of model, we can change the model ofˆusing (2) to one with the specified cluster picture; the values forˆ∕ andˆ∕ then follow from Theorem 2.17. By Lemma 3.4, ∕ = ∕ˆ∕ (−1) ord 2 ( ∕ ∕ˆ∕ ) , which gives the required values for . (2) First, note that if ′ is a different model for obtained by a Möbius transformation on the -coordinate (as in Definition 6.1), there is an isomorphism between Jacˆand Jacˆ′ that preserves the kernel of the corresponding isogeny. So, Jacˆand Jacˆ′ are isomorphic as abelian varieties with a principal polarisation, and hence,ˆ′ is isomorphic toˆby Torelli's theorem (see [25,Cor. 12.2]). We may therefore change the model of to ensure that it is centred and balanced (Theorem 6.5) and that it satisfies the conclusions of Theorem 6.6. This change of model does not change whether̂1 ∈ ×2 (Lemma 6.10) or the definition of (Lemma 6.9 for cases 2(a-f)). In particular, cases 2(b,c,e,f) will follow from cases 2(a,d). Note also that the cluster picture ofˆdepends on the choice of Richelot isogeny on , but not on the particular choice of C2D4 structure, so that cases 1 (b,d) will follow from cases 1 (a,c). By changing the model, it will thus suffice to establish the result for the following list of cases with the given simplifying hypotheses granted by Theorem 6.6; here , = ± are independent signs. These are proved in the sections indicated: We will use without further mention that in all of the above cases, the roots of and ofˆare integral, that is, ( ), ( ), (̂), (̂) ⩾ 0. This follows from Lemma 2.11 and, in cases 1× 1(b,c), 1× 1(b,c), I , , (b) and I ∼ (b), from the explicit formula in Lemma 8.3 below.
Notation 8.2. Throughout this section, for drawing cluster pictures, we will use the convention as in Theorem 7.1, that a cluster (other than ) with an index 0 means that the roots in it do not form a cluster. For example, when = 0, the cluster picture 0 means 0 .

Preliminary results
To control the cluster picture ofˆ, we will extensively use the following observations. Lemma 8.3. For a C2D4 curve with 1 , 2 , 3 ≠ 0, the roots of the Richelot dual curve arê ) , ) , ) .
(1, 3) Suppose that there is a cluster of depth that contains two roots of different colour: without loss of generality 2 and 3 . Then substituting 2 ≡ 3 mod into the expression for Δ gives Δ∕ ≡ ( 2 − 3 )( 1 − 2 )( 1 − 2 ) mod . For the pictures in (1), each term is a unit, so Δ∕ is a unit. For (3), either 1 or 1 is in the same cluster as 2 and 3 , so the corresponding term is ≡ 0 mod , and hence so is Δ∕ .
(2) In this case Δ∕ ≡ 2( 1 − 2 )( 3 − 1 )( 3 − 2 ) in the residue field, which is a unit. □ Lemma 8.7. Let ∕ℚ be a finite extension for an odd prime with residue field of size | | > 3. Let ∕ be a semistable C2D4 curve whose cluster picture has two clusters , ′ of size 3 with relative depth = 2 − and ′ = for some integers 0 ⩽ ⩽ ; we allow for the case =0 when ′ is not a cluster and is a cluster that is not contained in a cluster of size 4 or 5. Then admits another model with an identical cluster picture except for which = ′ = (and all colours, clusters, signs, Frobenius action on proper clusters, other relative depths and the depth of the full set of roots the same as for ).
Proof. This is essentially [15, Prop. 14.6(4)]. Write  for the set of roots for . If Galois swaps the two clusters, then they necessarily have the same depth, so there is nothing to prove. We may thus assume that is Galois stable. The expression on the right-hand side of Proposition 8.4 (1) is -rational and a perfect square in the residue field. Hence, a Galois element swaps the two clusters ofˆif and only if it swapŝ1 and̂1, if and only if it maps 1 to − 1 , if and only if it swaps the two clusters of .

Proof of Theorem 8.1: Toric dimension 1
Consider cases 1 (a), 1 2 (c) and 1× I ( ). By Lemma 8.5,ˆwill have either type 1 * or 1× * I * for some suitable * s. In particular, its cluster picture will have a cluster of size 2 or 4, but will not have two clusters of size 2 (Theorem 2.17).

8.4
Proof of Theorem 8.1: Toric dimension 2 Consider cases I , , (a,b), I ∼ (a,b), I ×I , I×I , U , , and U ∼ , . By Lemma 8.5,ˆwill have type I * , * , I * ×I * , I * × I * or U * * * , with some subscripts and signs. In particular, its cluster picture will have at least two clusters of size 2 or 4 (Theorem 2.17). Proof. Write + and − for the cyclic group on which Frobenius acts trivially and by multiplication by −1, respectively. The Néron component group of Jac ∕ is Φ = × , and similarly forˆ(see Remark 2.18). Passing to a quadratic ramified extension if necessary, we may assume that and are even.
We may now relabel the rootŝ↔̂, so that the three leftmost roots in the given pictures arê 1 ,̂2 and̂3, in some order, so that (̂−̂) = 0 for all , . Proposition 8.4 (4,5,6) gives Hence, the cluster picture must be First suppose that is of type I , , for some < . Without loss of generality, 2 ≡ 3 mod ∕2 . This gives Δ∕ ≡ ( 2 − 3 )( 1 − 2 )( 1 − 2 ) mod ∕2 , so that Δ has valuation exactly ∕2 and hence = 0. The above expressions and the restriction on the type then force the cluster picture ofˆto be 0 . Moreover, the sum of the depths of the sapphireturquoise twins is , and the explicit description of the roots in Lemma 8.3 shows that each depth is at least ∕2, so that each must be exactly ∕2. By All of the factors in these expressions are units except for 2 − 3 , which has valuation exactly 2 (as it is smaller than 2 ). Dividing through by 2 − 3 and then working modulo min( − 2 , 2 ) (so that also 2 ≡ 3 ), we get Hence, Δ Thus, the sign is the same as of the cluster { 2 , 3 } for , and has cluster picture with Frobenius action . Now suppose that the type is I , , or I ∼ . As (̂−̂) = 0, the same argument as for case I , , (a) shows that the cluster picture ofˆis 0 , 0 or 0 , in which the three leftmost roots have different colours, and similarly for the rightmost three (the fourth picture again cannot occur as it would yield 0 = 2 (Δ) − ( 1 ) = 2 + 2 > 0). The average valuation of -(i.e. of a sapphire minus a turquoise root) is higher than that of -, so at least one of the twins must consist of a sapphire and a turquoise root. The average valuation ofis the same as that of -, so both twins must be sapphire-turquoise, and the cluster picture is 2 -0 for some 0 ⩽ ⩽ 2 and some , > 0. For the two twins, from the valuations, we know that + = . By Theorem 2.17, the Tamagawa numbers satisfy Jacˆ∕ = and Jac ∕ = 2 ∕4, so that as the Richelot isogeny has degree 4, we necessarily have = 2 2 for some ∈ ℤ (Lemma 3.4). A little exercise in elementary number theory shows that as + = 1 and ⋅ = 2 , it follows that = = 2 .
Frobenius will swap the two twins (and hence the two clusters of size 3) if and only if it swaps the residues of̂2 and̂2. Working in the residue field, Proposition 8.4(2), Definition 2.13 and the cluster picture of tell us that Thus, Frobenius preserves the two twins ofˆwhen has type I +,+ , , I −,− , or I + ∼ , and swaps them for types I +,− , and I − ∼ . Together with Lemmata 8.5 and 8.7, this gives a model forˆwith the desired cluster picture. It follows that the cluster picture ofˆis 0 .
The only twins that can be swapped by Frobenius are the ones containing the ruby roots. This happens if and only if Frob(̂1 −̂1) =̂1 −̂1, which, by Proposition 8.4 (1), is equivalent to Frob( 1 ) = − 1 . It follows that the two twins forˆare swapped when is in case U ∼ , ( ), and not swapped in case U , , ( ). The sign forˆis determined by whether 1 2 3 Δ is a square (see Definition 2.13). As 1 ≡ 0, 2 ≡ 2 and 3 ≡ 3 , we have that 1  which forces the cluster picture ofˆto be 0 . The same argument as in cases I , , and I ∼ , shows that for case I * 2 × I * 2 , the cluster picture with Frobenius action ofˆis 0 , and for case I× I , it is + 0 . This completes the proof of Theorem 8.1.

ODD PLACES: ∕ ∕ = ∕
In this section, we will complete the proof of Theorem 7.1 by justifying the values it gives for ∕ and showing that ∕ ∕ = ∕ . Throughout we will use the division into cases as in Theorem 7.1. Then ∕ is as given in Theorem 7.1, and ∕ = ∕ ∕ .
Proof. Combine Lemmata 9.4, 9.5, 9.6, 9.7, 9.10, 9.11 and 9.12 below. □ The remainder of this section is devoted to the proof of this result. Throughout the section, ∕ℚ will be a finite extension for an odd prime and ∕ will be a centred semistable C2D4 curve with , Δ, 1 ≠ 0.

9.1
The value of ∕ ∕ Here we convert ∕ ∕ into Hilbert symbols. As both ∕ and ∕ are sensitive to the signs of twins, we first express these signs (defined via 2 , see Definition 2.13) in terms of our standard invariants from Definition 1.12.
Lemma 9.2. Let ∕ℚ be a finite extension for an odd prime and ∕ a centred semistable C2D4 curve with , 1 , Δ ≠ 0. For a twin cluster , 2 satisfies the following equalities: (2) In case 1 ( ) (       Proof. We will write for the residue field of , →̄for the reduction map tō, and ≡ for equality in the residue field (unless specified otherwise). In each case, we will show that, after a suitable scaling to make both sides units, the claimed identities hold over the residue field. The result then follows by Hensel's lemma.

The value of ∕
We now turn to the value of ∕ and show that ∕ ∕ = ∕ in all cases of Theorem 9.1. For convenience, we first recall some basic properties of Hilbert symbols. Recall that ( , ) = 1 if or is a square and whenever , are both units for odd places.   The second statement is then clear as either is a square, or both and − have even valuation. □ We will also make extensive use of the following identities in conjunction with Lemma 9.8(2). Lemma 9.9. Let be a field and ∕ a centred C2D4 curve with 1 ≠ 0. Then = 2( 2 + 1 )( 2 + 1 )( 3 + 1 )( 3 + 1 ) and Proof. Follows from direct computations using Definition 1.12. Proof. We will abbreviate cases 1× 1(a,b,c) as 1abc, 1× 1(a,b,c) as1abc and 2(a,b,c) as 2abc. We set = for these cases, as the two parameters will play an identical role. We also set = 0 for the cases 2(a,d).
From the cluster picture of , we find that The result follows by combining the following (see below for proof of †). We write 'both even' to mean ( , ) = 1 because ( ) and ( ) are even.

EVEN PLACES
In this section, we look at C2D4 curves with good ordinary reduction over 2-adic fields. Such curves admit a nice model -essentially, curves with good ordinary reduction turn out to be those with cluster picture with depth of each twin precisely (4). Theorem 10.3 then shows that Conjecture 1.14 holds for curves with this model and a specific Richelot isogeny. In the next section, we will show that the conjecture is independent of the choice of model and independent of the choice of the isogeny, and hence, that it holds for all curves with good ordinary reduction at 2-adic primes.
We begin with a preliminary lemma about 2-adic fields and Hilbert symbols.
Lemma 10.1. Let ∕ℚ 2 be a finite extension. Then is the quadratic unramified extension and ∈ such that 2 + Frob ∕ 2 is a unit, then Proof.
(i) Fix a set of representatives ⊇ {0, 1} of  ∕( ) and consider the equation for a given ∈  . Equating the coefficients of powers of , we must necessarily have 0 = 1 and 1 = ⋯ = −1 = 0 for = (2). The equation is then soluble if and only if 2 + ≡ mod is soluble. Hence, it is always soluble in the quadratic unramified extension of , but not in for a suitable choice of .
It follows that elements of the form = □ ⋅ (1 + 4 ) with ∈  are squares in the quadratic unramified extension of , and that some of these elements have ( √ ) ≠ . The set of such elements is a subgroup of × that properly contains ×2 , and hence, must contain all the elements ∈ such that ( √ )∕ is unramified. (ii) Follows from (i) and the fact that all units in are norms from any unramified extension.
for some , ∈ . Then 2 + Frob ∕ 2 = 2 2 + 2 2 , so that (ii) If ∕ has ordinary reduction and is the Richelot isogeny whose kernel is precisely the 2torsion points in the kernel of the reduction map, then the Richelot dual curveˆalso has good ordinary reduction. (ii) Take the model 2 = ( ) for ∕ given by (i). We will show thatˆhas a similar model, and hence also has good ordinary reduction by (i).
Using the fact that , have non-negative valuations (they satisfy a polynomial with unit leading term and integral coefficients) and the valuations of their pairwise differences, one readily checks that:  4 ) for some ∈  . Note that the quadratic twistˆ′ ofˆby (ˆ) does have good ordinary reduction by (i), and hence, its Jacobian has good reduction.
(iii) Take the model for ∕ given by (i). Relabelling the roots if necessary, by (ii), we may assume that the C2D4 structure is given by the , . We now need to adjust the model so that the claimed invariants are units.
As ( 1 − 1 ) ⩾ (4), the term 1 + 1 2 lies in  . Applying the translation to the -coordinate ↦ + 1 + 1 2 , we may thus assume that the C2D4 model is centred, that is, 1 = − 1 . Recall from Definition 6.3 that for ∈ { 1 1 , − 1 1 }, we have a Möbius transformation and model . We now proceed as in the proof of Theorem 6.6 to pick a suitable value for ∈  that gives a model with the required properties. None of these is the zero polynomial in ( ), as we now explain. Since 2 ≢ 3 and ↦ 2 is an automorphism of , we deduce that Thus, so long as ∈  avoids the residues of thē-roots of 1 ( ), 2 ( ) and 3 ( ) (at most 22 such) and the residues of −1∕ , −1∕ (at most 6 such), the required expressions will be units. □ ∈  × . Then Conjecture 1.14 holds for C/K.

DEFORMING C2D4 CURVES
As explained in §1.4, we will not attempt to prove other cases of Conjecture 1.14 by direct computation, as there are several hundred possible cluster pictures corresponding to semistable C2D4 curves. Instead, we will exploit the fact that we already have a good supply of C2D4 curves over number fields for which we have proved the 2-parity conjecture (through Conjecture 1.14 and Theorem 1.15) and use Lemma 1.19. For this, we will need to be able to approximate C2D4 curves over local fields by curves over number fields, which are well behaved at all other places. In this section, we prove two results that will let us do this (see Theorems 11.15 and 11.16). Roughly speaking, they say that: • a C2D4 curve ∕ can be approximated by a curve ′ ∕ such that Conjecture 1.14 holds for ∕ if and only if it holds for ′ ∕ , and moreover, holds for ′ ∕ for all ≠ ; • a curve with two C2D4 structures ∕ can be similarly approximated by ∕ admitting two C2D4 structures.
In §12, this will let us show that Conjecture 1.14 is independent of the choice of C2D4 model for a curve , and moreover, it holds with respect to one C2D4 structure if and only if it holds with respect to another (Theorems 12.1 and 12.3). These, in turn, will let us complete our proof of Theorem 1.16 on Conjecture 1.14 and deduce our main results on the 2-parity and parity conjectures in §13.
Lemma 11.2. For C2D4 curves over a local field of characteristic 0, the invariants 1 , 2 , 3 , Proof. This is clear for all the invariants except possibly and , as they are rational functions in the roots and the leading coefficient . For archimedean , =1, while is a locally constant function in the roots and by Lemmata 3.4 and 3.5 and the first paragraph of the proof of Theorem 5.2. For non-archimedean, the special fibre of the minimal regular model of ∕ (with Frobenius action) is locally constant, and hence, so is the deficiency term ∕ (cf. [15,Lemma 12.3]) and the local Tamagawa number Jac ∕ (cf. [5,Thm. 2.3]); the coefficients of the equation of the dual curveˆare continuous in the roots and in , so is also locally constant. The Galois representation (Jac ) ≅ 1 ( ∕ , ℤ )(1) is also locally constant ( [21, p.569 Proof. Clear as and are the same, and the terms in the Hilbert symbols in ∕ change by squares. □ a monic quartic with Gal( ( )) ⊆ 4 . There exists a monic quartic ( ) ∈  [ ] with Gal( ( )) ⊆ 4 such that (i) for each ∈ , the roots of ( ) are arbitrarily close to those of ( ) (with respect to an ordering that respects the 4 -action), (ii) for all ∉ , ( ) mod has no roots of multiplicity ⩾ 3.
Proof. The proof is the same as for Lemma 11.9, except that the parameters , , 1 , 2 , 3 , 4 ∈ (lying in  for ∉ ) are chosen as follows. First choose 1 and to be -adically close to 1, and for ∈ . Choose 3 and to be -adically close to 3, and for ∈ such that gcd( , 3 ) is supported on -this ensures that for primes ∉ that divide , ( 2 3 − 2 4 ) is a unit. Choose 2 to be -adically close to 2, for ∈ such that gcd( , 2 ) is supported on and 2 2 ≠ 2 3in particular, this ensures that for primes ∉ that divide , ( 2 2 − 2 4 ) is a unit. Finally, choose 4 to be -adically close to 4, for ∈ such that gcd( 2 2 − 2 3 , 4 ) is supported on -this ensures that for primes ∉ that do not divide either ( 2 2 − 2 4 ) or ( 2 3 − 2 4 ) is a unit. By Lemma 11.11, ( ) now satisfies (ii). Proof. Write ℎ ( ) = 2 + + . Using strong approximation (and the infinite place outside of ), pick ∈  that is -adically close to the for all ∈ and such that = ∏ ( + + ′ ) ≠ 0, the product taken over all pairs of roots of ( ) (including repeats). Let be the set of primes outside that divide ⋅ Disc( ( )).
Using strong approximation now pick ∈  so that (i) is -adically close to for all ∈ , and (ii) 2 + + mod is separable and coprime to ( ) mod for all ∈ .
The fact that the residue field at ∉ has size at least deg ( ) + 2 ensures that for each mod , there is always a polynomial over that satisfies (ii).
We can now take ( ) = 2 + + . Indeed, condition (ii) ensures that • if ( ) mod has a double root, then the roots of ℎ( ) mod are distinct from each other and from the roots of ( ) mod ; • the roots of ℎ( ) mod cannot both coincide with roots of ( ) mod for any ∉ : otherwise, we would have ≡ 0 mod , so that ∈ , which contradicts (ii); • if ℎ( ) mod has a double root (this would then be ≡ − ∕2 mod ) for ∉ , then it does not coincide with a root of ( ) mod . (1) 1 ( ) has cluster picture 0 , and (2) 2 ( ) has cluster picture 1 1 0 where , ( 2 , 2 ) and ( 3 , 3 ) denote the roots of , and , respectively.
Proof. Since ( ), ( ) ∈ [ ], the constraint on Gal(̄∕ ) means that its elements will either act trivially on the roots of ( ) ( ) or simultaneously swap the roots of ( ) and of ( ).
Write ∕ as 2 = ⋅ ℎ ( ) ( ) with ℎ ( ) a monic quadratic and ( ) a monic quartic with Gal( ( )) ⩽ 4 given by the C2D4 structure. In the case that is non-archimedean, we may assume that ℎ ( ), ( ) ∈  [ ]: otherwise scale by a suitable totally positive element of whose only prime factor is (this exists as has finite order in the class group).
Let be the set consisting of , all real places other than ′ , primes above 2 and all primes with residue field of size < 23. For ∈ ⧵ { }, define C2D4 curves ∶ 2 = ℎ ( ) ( ) over for quadratic ℎ ( ) and quartic ( ) as follows: • For |∞, let = −1 and ℎ ( ) = ( − 1), ( ) = ∏ 5 =2 ( − ), so has picture . the roots of each quadratic are monochromatic. The condition in (2) is equivalent to a factorisation into a quadratic ( ) with the ruby roots and a quartic ( ) whose Galois group is contained in 4 . Thus, to prove the theorem, it will suffice to construct ∕ so that • it satisfies (ii), (iv) and (iii), avoiding the cluster picture in case (1), and • in case (1) Indeed, such a curve will automatically admit two C2D4 structures that satisfy (i). The construction of ∕ follows exactly as in the proof of Theorem 11.15, except that the use of Lemmata 11.9 and 11.13 in the penultimate paragraph is replaced by two applications of Lemma 11.13 in case (1) and by Lemmata 11.12 and 11.13 in case (2), and that the step in the final paragraph is not relevant here. □

11.7
Making the terms  , , ≠ Recall that we will eventually need to address the special cases when , 1 , or Δ is 0. Here we record the methods to make small perturbations to the given C2D4 model to make these invariants non-zero.
Lemma 11.17. Let be a local field of characteristic 0 and ∕ a centred C2D4 curve. Then there is a 0 ∈ arbitrarily close to 0 such that for all ∈ sufficiently close to 0 the model has  ≠ 0.
Proof. It suffices to find one value of 0 close to 0 such that 0 has  ≠ 0, since ( ) is continuous as a function of . By definition  = 1 2 3 2 3 ( 2 + 3 )( 2 2 + 3 3 )(̂2 3 +̂3 2 ). As the individual factors are rational functions in , it suffices to prove that none of them are identically zero for ∈̄.

MAIN LOCAL THEOREM: GENERAL CASE
We now return to the proof of Conjecture 1.14. Proof. Write = as the completion of some number field at a place , which also has a complex place ′ ≠ .

Changing the model
We may change the C2D4 model by scaling the -coordinate (this changes the leading term by a square), as this does not affect any of the Hilbert symbols in ∕ and hence the validity of Conjecture 1.14 for ∕ . By Remark 6.2, we may therefore assume that the model ′ is for some ∈ GL 2 ( ). Since and −1 are continuous, by Lemma 11.19, we may moreover assume that 1 ≠ 0 for both and ′ .
By Theorems 11.15 and 4.1, there is a C2D4 curvẽdefined over which is arithmetically close to over and for which Conjecture 1.14 holds at all places of . Moreover, by continuity of −1 , we can pick it to be -adically sufficiently close to so that (̃) −1 is arithmetically close to . Now use continuity (Lemmata 11.3 and 11.4) and strong approximation to pick ′ ∈ GL 2 ( ) such that (i) ′ is -adically close to −1 , so that̃= (̃) ′ is arithmetically close to , and (ii) ′ is -adically close to the identity at all places ≠ , ′ that are either archimedean or wherẽ has bad reduction, so that̃is arithmetically close to (̃) at these places.
To summarise, we have now replaced the pair of curves , defined over by a pair̃,d efined over such that • and̃are arithmetically close over , • and̃are arithmetically close over , • Conjecture 1.14 holds for̃at all places of , • Conjecture 1.14 holds for̃at all places ≠ , ′ of that are archimedean or where has bad reduction, and hence by Theorem 4.1 at all places ≠ of . By Theorem 1.15, the 2-parity conjecture holds for̃∕ . Sincẽis another model for̃, the 2-parity conjecture also holds for̃∕ . By Lemma 1.19, it follows that Conjecture 1.14 must also hold for̃at the remaining place . Since this curve is arithmetically close to over = , the conjecture also holds for ∕ , as required. □ (2) = 2, ∕ has good ordinary reduction and the kernel of the associated Richelot isogeny on the Jacobian is precisely the kernel of the reduction map on 2-torsion points.

Finite places
Proof. We consider the cases of odd and even residue characteristic independently. By Lemma 11.19, we may assume that 1 ≠ 0. By Lemma 3.7, we may assume that the residue field of is sufficiently large. The result now follows from Theorems 4.1, 6.6 and 12.1 for odd, and from Proposition 10.2 and Theorems 10.3 and 12.1 for = 2. □

12.3
Changing the isogeny Theorem 12.3. Let be a non-archimedean local field of characteristic 0. Let ∶ 2 = ( ) be a curve over that admits two C2D4 structures, (1) and (2) , both of which have , Δ ≠ 0 and such that Conjecture 1.14 holds for (1) . Suppose that one of the following two conditions holds: (1) • the second colouring is obtained from the first by relabelling colours, and • Gal( ) preserves colours; or (2) • both colouring have the same ruby roots, and • Gal( ) acts on the sapphire and turquoise roots as a subgroup of 4 .
Proof for a finite extension of ℚ 2 . Since the validity of Conjecture 1.14 is unchanged by going to an unramified extension of odd degree (Lemma 3.7), we may assume that [ ∶ ℚ 2 ] > 1. Pick a number field that has a prime above 2 with completion ≃ , and such that has no other primes above 2 and has a complex place. (To see that such a field exists, pick a primitive generator for ∕ℚ 2 and approximate its minimal polynomial by a polynomial ( ) ∈ ℚ[ ] that has at least two complex roots; then = ℚ[ ]∕ ( ) has the required property.) Over local fields, a small perturbation to the coefficients of a separable polynomial does not change its Galois group, so by Lemma 11.19, we may assume that both curves have 1 ≠ 0. By Theorems 11.16 and 4.1, there is a curvẽ∕ that admits two C2D4 structures̃( 1) and̃( 2) such that̃( ) is close to ( ) and such that Conjecture 1.14 holds for both̃( 1) and̃( 2) at all places ≠ . In particular, Conjecture 1.14 holds for̃( 1) at all places of , and hence, the 2-parity conjecture holds for̃(Theorem 1.15). It thus also holds for̃ ( 2) , and thus, by Lemma 1.19, it follows that Conjecture 1.14 must hold for̃( 2) ∕ , and hence for (2) ∕ . □ Proof. The curve ∶ 2 = ( ) has all its Weierstrass points defined over . As Gal( ) is trivial, we can repeatedly apply Theorem 12.3 (and Lemma 11.21) to the given C2D4 structure (1) to change it to the standard C2D4 structure (0) ∈  2 4 . Conjecture 1.14 holds for (0) (Theorem 10.5), and hence for (1) as well. □

Changing the isogeny (continued)
Proof of Theorem 12.3 for archimedean or a finite extension of ℚ , odd. Let be a number field with a prime such that ≃ and has a complex place ′ ≠ . By Theorem 12.4, Conjecture 1.14 holds for all curves over 2-adic fields that lie in  , irrespectively of the choice of the C2D4 structure. The proof now follows verbatim as the third paragraph of the proof of the case when is an extension of ℚ 2 . □

12.6
Proof of Theorem 1.16 Theorem  Proof. Write the curve as ∶ 2 = ( ) and consider the colouring of the roots of ( ) given by the C2D4 structure, (1) . Observe that (1) if Gal( ) preserves colours, then admits two other C2D4 structures obtained from the original one by relabelling the colours; and (2) if Gal( ) acts as a subgroup of 4 on the sapphire and turquoise roots, then admits two other C2D4 structures obtained from the original one by changing the colouring of sapphire and turquoise roots.
Let (2) be any one of these structures. By Lemma 11.21, we may assume that (2) also has , Δ ≠ 0. By Theorem 12.3, it then suffices to prove the result for (2) . We now show that through repeated use of (1) and (2), we can reduce the problem to one already covered by Theorems 4.1 and 12.2.
Complex places: The result is covered by Theorem 4.1.
Real places: We may assume that if ( ) has a real root, then < 0: indeed, by Theorem 12.1, we can use a change of model given by ↦ 1 − for a suitable ∈ ℝ to make the leading term negative. If ( ) has six real roots, then a repeated use of (1) and (2) brings it to the picture . If ( ) has four real roots, then by (1), we may assume that the complex roots are ruby, and then, by (2) that the picture is with 1 =̄1. If ( ) has two real roots, then by (1), we may assume that these are ruby, and then by (2) that the picture is with 2 =̄2 and 3 =̄3, and then by (1) again that the picture is instead with 1 =̄1 and 3 =̄3.
Odd primes: By Theorems 6.5 and 12.1, we may assume that the cluster picture of ∕ is balanced (using Lemma 3.7(5) to enlarge | | if necessary). If the reduction has type 2 or 1 (in the sense of Theorem 2.17), the result follows from Theorem 12.2. Otherwise, its cluster picture is one of the ones given below. Applying steps (1) and (2) as indicated above the arrows reduces the problem to one covered by Theorem 12.2.
2-adic primes: By Proposition 10.2(i) and Theorem 12.1, we may assume that the cluster picture of has three twins of depth (4), and that the depth of the cluster containing all six roots is 0 (using Lemma 3.7(5) to first enlarge | | if necessary). The result follows as for the case of Type U , , above. □

GLOBAL RESULTS
We now complete the proofs of the theorems given in the introduction. Proof. If ≃ ∏ or ≃ Res ∕ , then the condition on the 2-torsion field ensures that the elliptic curves ∕ or ∕ all admit a 2-isogeny. By [13,Thm. 5.8], the 2-parity conjecture holds for ∕ (respectively, ∕ ). As the 2-parity conjecture is compatible with products and with Weil restriction of scalars (as both root numbers and ∞ -Selmer ranks are), it also holds for ∕ . □ Theorem 13.2. The 2-parity conjecture holds for all C2D4 curves over number fields ∕ with Δ = 0.
Proof. By [31,Def. 8.2.4 and Prop. 8.3.1], = Jac has an isogeny of degree 4 to an abelian variety that is either a product of two elliptic curves or the Weil restriction of an elliptic curve from a quadratic extension. By hypothesis, Gal( ( [2])∕ ) is a 2-group, and hence so is Gal( ( [2 ])∕ ). It follows that Gal( ( [2])∕ ) is also a 2-group. The result now follows by Proposition 13.1, since the 2-parity conjecture is compatible with isogenies (as both root numbers and ∞ -Selmer ranks are invariant under isogenies). □ Theorem 13.3. The 2-parity conjecture holds for all principally polarised abelian surfaces over number fields ∕ such that Gal( ( [2])∕ ) is a 2-group that are either • the Jacobian of a semistable genus 2 curve with good ordinary reduction at primes above 2, or • not isomorphic to the Jacobian of a genus 2 curve.
Proof. By [17,Thm. 3.1], is either a product of two elliptic curves, the Weil restriction of an elliptic curve from a quadratic field extension or is the Jacobian of a genus 2 curve ∕ . By Proposition 13.1 and the hypothesis on the 2-torsion field, we may assume that = Jac for a C2D4 curve ∕ . By Theorem 13.2 and Lemma 11.17, we may also assume that , Δ ≠ 0. Proof. Let ∕ be a C2D4 curve over a local field of characteristic 0 with  ≠ 0. Let be a number field with a place , such that ≅ , and some other place ′ that is archimedean. By Lemma 11.19, we may assume that 1 ≠ 0. By Theorems 11.15 and 12.5, we can find a C2D4 curve ′ over such that and ′ are arithmetically close over and such that Conjecture 1.14 holds for ∕ for all places ≠ . By assumption, the 2-parity conjecture holds for ′ ∕ , so by Lemma 1.19, Conjecture 1.14 holds for ∕ . It follows by Lemma 11.4 that Conjecture 1.14 also holds for ∕ . □

A.1 Statement of the result
Let be a finite extension of ℚ 2 and ∕ a principally polarised abelian variety of dimension g, with good ordinary reduction. Let be a maximal isotropic subspace of [2] (for the Weil pairing associated to the principal polarisation), stable under the action of the absolute Galois group Gal(̄∕ ). Let ∶ → be the -isogeny with kernel , so that is principally polarised also, has good ordinary reduction, and (after identifying and with their duals) the dual isogenŷ ∶ → satisfies •̂=̂• = [2].
Let 1 (̄) denote the kernel of reduction on . The aim of the Appendix is to prove the following result, whose proof we give in §A.3 after reviewing endomorphisms of the formal multiplicative group.

A.2 Endomorphisms of the formal multiplicative group
Again, let be a finite extension of ℚ 2 , and let denote the completion of the maximal unramified extension of . Let  be the ring of integers of , so that  is a complete discrete valuation ring, whose normalised valuation restricts to that of . Letˆdenote the formal multiplicative group over . In general, given formal group laws  and  over  of dimension g, and a homomorphism from  to , we denote by ( ) the Jacobian of . That is, is an -tuple of power series in g variables = ( 1 , … , g ), coefficients in , and ( ) ∈ M g () is an g × g matrix such that Proof. The result for general g follows formally from the case g = 1, which is standard, although we provide the proof for convenience. The formal logarithm gives an isomorphism fromˆto the formal additive groupˆover , and the endomorphisms of the latter are given by ( ) = for ∈ . Thus, one sees that the endomorphisms ofˆover  are exactly those of the form ( ) = + ( − 1) 2 ∕2 + ( − 1)( − 2) 3 ∕3! + ⋯ for those ∈ such that each coefficient of ( ) is in . Considering the coefficients of 2 for varying , one sees easily that this is equivalent to ∈ ℤ 2 , from which the result follows. □ Now let denote the maximal ideal in . Letting 1 ( ) denote the group of units in  reducing to 1 in the residue field ∕ , the map ( 1 , … , g ) ↦ ( 1 − 1, … , g − 1) gives an isomorphism from 1 ( ) g to g with the group structure on the latter coming from the formal group lawˆg . Any endomorphism ∈ End  (ˆg ) induces via this isomorphism an endomorphism of 1 ( ) g . We denote by 1 ( ) g [ ] the kernel of this map. Lemma A.3. Let ∈ End  (ˆg ) and suppose that there is ∈ End  (ˆg ) such that • = [2] (here [2] denotes the multiplication-by-2 map onˆg ). Then 1 ( ) g [ ], being contained in 1 ( ) g [2] = {±1} g , is a finite-dimensional 2 -vector space and we have dim 2 1 ( ) g [ ] = ord 2 det ( ).
Proof. Let = ( ) ∈ g (ℤ 2 ). By properties of Smith Normal Form, we can find invertible matrices and in g (ℤ 2 ) such that = where is a diagonal matrix whose entries are powers of 2. On the other hand, ( ) ( ) is twice the identity matrix. Thus, In particular, each coefficient of ( ) is divisible by 2, yet 1 2 ( ) has determinant a 2-adic unit. If one of the entries of was divisible by 4, then 2 would divide each entry of some row of 1 2 ( ), and hence its determinant, a contradiction. We deduce that each entry of is either 1 or 2. Moreover, the matrices and correspond to automorphisms ofˆg under Lemma A.2 and since we are only interested in the size of the kernel of , we may replace with the endomorphism corresponding to (by construction, we also have ord 2 det ( ) = ord 2 det ). However, as is diagonal with entries either 1 or 2, the endomorphism of 1 ( ) g induced by is just the identity on each factor where has a 1 on the diagonal, and the map ↦ 2 on each factor where has a 2 on the diagonal. The 2 -dimension of the kernel of this map is just the number of diagonal entries of equal to 2, which is equal to ord 2 det . □

A.3 Proof of Theorem A.1
We keep the notation of § .1 and § .2, so that, in particular, let ∕ be a principally polarised abelian variety of dimension g with good ordinary reduction, let ∕ be isogenous to via , and consider the auxiliary isogeny ∶ → such that • = [2]. Let be the normalised valuation on (which extends that on ), denote the residue field of , and let ( ∕ℚ 2 ) be the ramification index of over ℚ 2 . Let  and  be the dimension g formal group laws over the ring of integers  of associated to and , respectively. Then induces an element of Hom  ( ,  ) which, by an abuse of notation, we also denote by . Similarly, we obtain ∈ Hom  ( ,  ) and we have • = [2]. Since and have good ordinary reduction, over  (the ring of integers of ), there is an isomorphism from  toˆg , and similarly an isomorphism from  toˆg (see [24,Lemma 4.27] for more details). We thus obtain elements ′ ∶= −1 and ′ ∶= −1 of End  (ˆg ) whose composition is multiplication by 2. Moreover, since and are isomorphisms, ( ) and ( ) are invertible matrices in (). In particular, the determinants of ( ) and ( ) are units in . Thus, (det ( )) = (det ( ′ )) = ( ∕ℚ 2 ) ord 2 det ( ′ ). Let̄be an algebraic closure of and let 1 (̄) [2] denote the points in (̄) [2] reducing to the identity under the reduction map intō(the algebraic closure of ). Then since has good ordinary reduction, 1 (̄) [2] has size 2 g . On the other hand, the points in 1 ( ) correspond (1) has 2-power order; or (2) there is ⊲ with ∕ ≅ ⋊ 2 for an odd prime, ⩾ 0, and 2 acting faithfully on .
Proof. We proceed by induction on | |. Solomon's induction theorem expresses as an integral linear combination of Ind for some hyperelementary < . As induction is transitive, we may assume that is hyperelementary. (Recall that a group is hyperelementary if ≃ ⋊ for a -group and a cyclic group of order prime to .) If has a non-trivial odd order quotient, then it has a -quotient for some odd prime , and we are done by (2). Otherwise, = ⋊ for some odd and a 2-group . If = 1, we are done by (1). By passing to a quotient if necessary, we may assume that is prime and, moreover, that acts faithfully. Then we are done by (2). □ Recall that for a prime , we define the dual ∞ -Selmer group, This is a ℚ -vector space whose dimension is the Mordell-Weil rank of ∕ plus the number of copies of ℚ ∕ℤ in X( ∕ ). If X is finite, this is equivalent to the parity conjecture.
If the -parity conjecture holds for ∕ for all ⊊ ⊂ , then it holds for ∕ . Proof.
Since -parity holds over 2 and over by assumption, it holds for the twists of by