A user's guide to the local arithmetic of hyperelliptic curves

A new approach has been recently developed to study the arithmetic of hyperelliptic curves y2=f(x)$y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of f$f$ . Since its introduction, numerous papers have used this machinery of ‘cluster pictures’ to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self‐contained fashion, complemented by an abundance of examples.


INTRODUCTION
In this paper, we provide a summary of a recently developed approach to understanding the local arithmetic of hyperelliptic curves. This approach revolves around the theory of 'clusters', and enables one to read off many local arithmetic invariants of hyperelliptic curves from explicit equations 2 = ( ). The paper is meant to serve as a user's guide: our aim has been to make it accessible to mathematicians interested in applications outside of local arithmetic geometry, or who may wish to compute local invariants without having to decipher the theoretical background.
Throughout this article, will be a local field of odd residue characteristic and ∕ a hyperelliptic curve given by where ∈ [ ] is separable, deg( ) = 2g + 1 or 2g + 2 and g ⩾ 2.

How to use this guide
The article is structured as follows. We begin in Section 2 by declaring some general notation which will be used throughout, and proceed to give some background theory on cluster pictures and BY trees in Sections 3 and 4, respectively. Cluster pictures will be critical background for all sections of the article; BY trees will be used in Sections 10, 15, 17 and the Appendix. From there on, each section will be self-contained and independent of the other sections. This will allow a reader who is concerned with just one topic (Galois representations, say) to be able to learn everything they need by reading just the background theory in Sections 3 and 4 and the relevant section (in our example, Section 11).
From Section 5 onwards, each section will consist of two parts: the first stating the relevant theorems, and the second providing examples illustrating the theorems. None of the theorems are original (apart from Theorem A.6, whose proof is given in the Appendix) and we give no proofs; each section has references at the end where the interested reader can find proofs and more general statements of the theorems.

Related work
The key references for the present work are [3,4,9,10,14,15,20]. We have made a blanket assumption that is a local field; this is often unnecessarily restrictive, and many results hold for complete discretely valued fields. The reference [9] also discusses a number of topics that we have omitted, in particular how to use clusters to check whether a curve is deficient, how one may perturb ( ) without changing the standard local invariants, and how to classify semistable hyperelliptic curves in a given genus. As many of our examples will illustrate, the method of cluster pictures is very convenient for computations. However, it can also be used for more theoretical purposes: for instance, one can work explicitly with families of hyperelliptic curves for which the genus becomes arbitrarily large (see, for example, [1,7]), or prove general results for curves of a given genus by a complete caseby-case analysis of cluster pictures (see, for example, [13]).
We would like to mention some alternative techniques that have been recently developed for investigating similar topics. In [6,16,[21][22][23], the authors determine different kinds of models, the conductor exponent, the local -factor, compare the Artin conductor to the minimal discriminant and compute a basis of the integral differentials. In arbitrary residue characteristic (including 2), but under some technical assumptions, [8,12,20] determine the minimal regular model with normal crossings, a basis of integral differentials, reduction types, conductor and action of the inertia group on the -adic representation.

Implementation
We have implemented many of the methods described in this guide as a package using the Sage-Math computer algebra system [24]. The package is available online at [2]. This package includes implementations of cluster pictures and BY trees as abstract objects, which it can also plot. Given a hyperelliptic curve, the implementation determines its associated cluster picture and BY tree. It also determines the Tamagawa number, root number, reduction type, minimal discriminant and dual graph of the minimal regular model, as described in this article. We have also computed cluster pictures for all elliptic curves over ℚ and number fields, and all genus 2 curves present in the L-Functions and Modular Forms Database [19]. The latter is incorporated in the LMFDB homepages of curves.

NOTATION
Here we set out the notation that will be used throughout the paper.
Formally by a hyperelliptic curve we mean the smooth projective curve associated to 2 = ( ), equivalently the gluing of the pair of affine patches 2 = ( ) and 2 = 2g+2 ( 1 ) along the maps = 1 and = g+1 , where ∈ [ ] is separable, and deg( ) ⩾ 5. We will not consider double covers of general conics. We fix the following notation associated to fields and hyperelliptic curves.  We will say is semistable if has semistable reduction. Similarly is tame if acquires semistable reduction over a tame extension of . If > 2g + 1, is always tame, see Remark 5.7.

CLUSTERS Definition 3.1 (Clusters and cluster pictures).
A cluster is a non-empty subset ⊆  of the form = ∩  for some disc = { ∈̄| ( − ) ⩾ } for some ∈̄and ∈ ℚ. For a cluster with | | > 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 .
(ii) If ( ) mod has a double root and no other repeated roots, then the cluster picture has a twin and̄has a node. Generally, for semistable curves, twins contribute nodes to the special fibre of the stable model. (iii) The normalisation of̄is obtained by removing the maximal square factor in̄( ), so the new roots are in 1:1 correspondence with the odd clusters. Explicitly, it is the hyperelliptic curve given by 2 = ∏ ∈ ( −̄). For example, for the curve in (i), the normalisation is given by 2 = ( − 1)( − 2)( − 3). (iv) When  is übereven, the normalisation of̄is 2 = 1, which is a union of two lines. Generally, for semistable curves, übereven clusters contribute pairs of ℙ 1 s to the special fibre of both semistable and regular models of ∕ℚ nr . (v) Suppose that  = {1, 2, , 2 , 3 , 4 } so the cluster picture is for > 3. Applying the change of variable ′ = 1 gives a curve whose cluster picture is . Generally, changing the model can convert twins to cotwins and vice versa, and the number of twins plus cotwins is model independent.
(vi) For a curve as in (ii), the node on̄is split if and only if ∏ ∉ (̄−̄) is a square in . Equivalently, if and only if (Frob) = +1. Generally, keeps track of whether the nodes are split or non-split and similar data. (i) Suppose that ( ) mod has distinct roots, equivalently that the cluster picture of is . In this case,  = ( ). Here when  is even, has good reduction and when  is odd it is a quadratic twist of such a curve.
(ii) Suppose that ( ) mod has a repeated root of multiplicity 5 and the corresponding roots inQ are equidistant with distance ( − ) = , equivalently the cluster picture of is .
In general, for any proper cluster the above change of variable will give an integral equation for of the form ℎ( ) where ( ) + = and ℎ( ) is integral. When ∈ ℤ, ∈ ℚ and ∈ 2ℤ, the substitution = ′ 2 gives an equation for ∕ℚ whose reduction is of the form When is principal, this is a curve over of genus at least 1.
The general case is more subtle. Roughly, for a proper cluster , the denominator of̃is related to the inertia action oñand to the inertia action by geometric automorphisms on the reduced curve associated to à la Example 3.8.

BY TREES
Definition 4.1 (BY tree). A BY tree is a finite tree with a genus function g ∶ ( ) → ℤ ⩾0 on vertices, a length function ∶ ( ) → ℝ >0 on edges, and a 2-colouring blue/yellow on vertices and edges such that (1) yellow vertices have genus 0, degree ⩾ 3, and only yellow incident edges; (2) blue vertices of genus 0 have at least one yellow incident edge; (3) at every vertex, 2g( ) + 2 ≥ # blue incident edges at .
Note that all leaves (vertices of degree 1) are necessarily blue.

Notation 4.2.
In diagrams, yellow edges are drawn squiggly ( ) and yellow vertices hollow ( ) for the benefit of viewing them in black and white. We write the genus of a blue vertex inside the vertex ( ); we omit it for blue vertices with genus 0. We write the length of edges next to them. Definition 4.3. The BY tree associated to is given by: • one vertex for every proper cluster , coloured yellow if is übereven and blue otherwise; • for every pair ′ < with ′ proper, link ′ and with an edge, yellow of length 2 ′ if ′ is even and blue of length ′ if ′ is odd; • if  has size 2g + 2 and is a union of two proper children, remove  and merge the two remaining edges (adding their lengths); • if  has size 2g + 2 and has a child of size 2g + 1, remove  and the edge between  and ; • the genus g( ) of a blue vertex is defined so that |̃| = 2g( ) + 2 or 2g( ) + 1. Equivalently, is a collection of signs ( ) ∈ {±1} and ( ) ∈ {±1} for every yellow vertex and yellow edge, such that ( ) = ( ) whenever ends at . Isomorphisms are composed by the cocycle rule ) .
An automorphism of is an isomorphism from to itself. Notation 4.6. We draw arrows between edges and signs above yellow components to represent automorphisms.
Remark 4.7. For semistable curves, inertia maps to the identity in Aut , that is = id and ( ) = +1 for all ∈ and all yellow components . Example 4.9. Consider the cluster picture from Example 3.6. There are four proper clusters , , 1 and 2 so the BY tree has four vertices  , , 1 , 2 , where only is yellow since is übereven. There are three yellow edges corresponding to the three even children < , 1 < , 2 < , of length 2 × 5, 2 × 3 2 , 2 × 3, respectively. is not a square in 11 ( 7 ), so 2 (Frob) = −1. In terms of the BY tree, Frob permutes the three edges cyclicly. Here the yellow components are the three edges  1 ,  2 ,  4 , and Frob (  1 ) = Frob (  4 ) = +1, while Frob (  2 ) = −1. Note that these all have the same BY tree: Generally, the BY tree is model independent.

REDUCTION TYPE
In this section, we explain how to read off information about the reduction of both and its Jacobian from the cluster picture of . (1) The field extension ()∕ given by adjoining the roots of ( ) has ramification degree at most 2.

(2) Every proper cluster is invariant under the action of the inertia group .
(3) Every principal cluster has ∈ ℤ and ∈ 2ℤ.
Remark 5.2. It follows from Theorem 5.1 that is semistable over any ramified quadratic extension of (). (1) The field extension ()∕ is unramified.
A consequence of Theorems 5.3 and 5.4 is the following criterion for potentially good reduction.
Theorem 5.5 (Potentially good reduction of the curve or the Jacobian).
• The curve has potentially good reduction if and only if every proper cluster has size at least 2g + 1.
• The Jacobian, Jac , has potentially good reduction if and only if every cluster ≠  is odd.
Theorem 5.6 (Potential toric rank of the Jacobian).
• The potential toric rank of Jac is equal to the number of even non-übereven clusters ≠ , less 1 if  is übereven. • The Jacobian, Jac , has potentially totally toric reduction if and only if every cluster has at most 2 odd children.
Remark 5.7 (Tame reduction). The curve , or equivalently Jac , has tame reduction (semistable after tamely ramified extension) if and only if ()∕ is tamely ramified. In particular, this is always the case if > 2g + 1 since then the wild inertia group acts trivially on the roots of the (degree ⩽ 2g + 2) polynomial ( ).
Remark 5.9. Any hyperelliptic curve ∶ 2 = ( ) with the same cluster picture as the one in Example 5.8 (same depths, all proper clusters inertia invariant) and such that ( ) has unit leading coefficient, is necessarily also semistable with totally toric reduction, by the same argument.
Example 5.10. Consider the genus 2 hyperelliptic curve ∶ 2 = 6 − 27 over ℚ 3 . Its cluster picture is for a fixed primitive third root of unity 3 . The non-principal cluster  has depth 1 2 , whilst the principal clusters 1 and 2 each have depth 1. We find: • is not semistable since the action of inertia swaps 1 and 2 ; • does not have potentially good reduction, since 1 and 2 are both proper clusters of size < 2g + 1. On the other hand, Jac does have potentially good reduction since 1 and 2 are odd; • has tame reduction since ℚ 3 () = ℚ 3 ( √ 3, 3 ) has ramification degree 2 over ℚ 3 . In fact, the minimal degree extension over which is semistable is 4, realised by any totally ramified extension of this degree. To see this, note that the inertia group acts on the proper clusters through its unique order 2 quotient, whilst for = 1, 2, we have ∈ ℤ and = 3 ⋅ 1 + 3 ⋅  = 9∕2, so that satisfies the semistability criterion (Theorem 5.1) over some ∕ℚ 3 if and only if the ramification degree of this extension is divisible by 4.

SPECIAL FIBRE (SEMISTABLE CASE)
In this section, assuming that ∕ is semistable and that  is principal, we describe the special fibre of the minimal regular model of over  nr . The case where  is not principal is dealt with in [9, Section 8].
Definition 6.1 (Leading terms and reduction maps). For a principal cluster , define ∈̄× and red ∶ + ̄→̄by For any cluster ′ < , we define red ( ′ ) to be red ( ) for any choice of ∈ ′ . Theorem 6.2 (Components). The special fibre  min contains connected components Γ corresponding to principal clusters , given by the equations This component is irreducible when is non-übereven but splits into a pair of irreducible components Γ + , Γ − otherwise (we write Γ + = Γ − = Γ in the non-übereven case). These components are linked by chains of ℙ 1 s as described in Theorem 6.3.

Theorem 6.3 (Links).
The chains of ℙ 1 s linking the irreducible components of Theorem 6.2 arise in exactly one of the following four ways. If ′ < with both clusters principal and ′ is odd, we have a chain containing 1 2 ′ − 1 components, linking Γ to Γ ′ . If ′ < with both clusters principal and ′ even, we have two chains containing ′ − 1 components each, one linking Γ + to Γ + ′ and the other Γ − to Γ − ′ . If < with principal and a twin, we have a chain containing 2 − 1 components, linking Γ + to Γ − . When these conditions are satisfied, the reduction is given by If = , then the reduction of ( , ) ∈ ( nr ) lies on Γ  if and only if either (6.7) holds, or ( −  ) < . In the former case, the reduction is given by (6.8), whilst in the latter case ( , ) reduces to one of the points at infinity on Γ  . † Example 6.9. Consider the genus 2 curve ∶ 2 = (( + 1) 2 − 5)( + 4)( − 6) over ℚ 5 with associated cluster picture Picking  = 0, we have red  ( ) = mod and  = 1 ∈̄× 5 . The special fibre of the minimal regular model has a component coming from the unique principal cluster  given by the equation a genus 0 curve with a single node at ( , ) = (−1, 0). For the twin 1 , we have 2 1 − 1 = 0 so that 1 contributes no components (rather, it corresponds to the node on Γ  ). On the other hand, the twin 2 gives rise to a chain of 2 2 − 1 = 1 projective lines from Γ  to itself, as pictured below.
A point ( , ) ∈ (ℚ nr 5 ) reduces to a point on Γ  if and only if either ∉ ℤ nr 5 , in which case it reduces to the unique point at infinity on Γ  , or ∈ ℤ nr 5 and ≢ ±1 mod 5. Since  = 0, for points satisfying the second condition the reduction map is given by ( , ) ↦ (̄,̄(̄− 1) −1 ). Example 6.10. Consider ∶ 2 = ( 4 − 8 )(( + 1) 2 − 2 )(( − 1) 2 − ) over ℚ , with associated cluster picture Then  and are the only principal clusters. Moreover,  is übereven. Taking  = = 0, we get associated components of  min : The parent-child relation <  gives rise to two chains of length = 1, one linking Γ +  with Γ , and the other linking Γ −  with Γ . The twin 1 gives rise to a chain of length 2 The twin 2 has 2 2 − 1 = 0 so contributes a chain of length 0 from Γ −  to Γ +  , which is to be interpreted as a point of intersection between these two curves. The configuration of the components of the special fibre is shown below. Finally, since both  and are -stable, and

MINIMAL REGULAR MODEL (SEMISTABLE CASE)
Throughout this section, we assume that is semistable. We also assume for simplicity that all proper clusters have integral depth, and that there is no cluster ≠  of size 2g + 1.
The point is called a centre of , and is called its depth. The parent disc ( ) of is the disc with the same centre and depth − 1. We also write ( ) = ( ) + ∑ ∈ min{ , ( − )}, and ( ) ∈ {0, 1} for the parity of ( ). We write () for the smallest disc containing . An integral disc is called valid when ⊆ () and #( ∩ ) ⩾ 2.

Construction of a regular model  over 
Firstly, for each valid disc , we let ( ) ∈  nr [ ] denote the polynomial ( ) = − ( ) ( + ). We set  to be the subscheme of 2  nr cut out by 2 = ( ) ( ).
We let  • denote the open subscheme of  formed by removing all the points in the special fibre corresponding to repeated roots of the reduction of (viewed as points on  with = 0). Next, for the maximal valid disc = () we let g ( ) ∈  nr [ ] denote the polynomial g ( ) = deg( ) (1∕ ). We set  to be the subscheme of 2  nr cut out by 2 Remark 7.4. The regular model  disc above is not minimal in general: discs with ( ) = 1 produce ℙ 1 s in the special fibre with multiplicity 2 and self-intersection −1. Blowing down these components yields the minimal regular model. Remark 7.5. In the construction of  disc in [9, Proposition 5.5] for general semistable , the scheme  • is defined by removing from the special fibre of  all points corresponding to the maximal valid subdiscs of . Under our extra assumptions, this is equivalent to the reduction of having a repeated root at this point. This is untrue when has a twin of half-integral depth; see Example 7.7. Example 7.6. Consider ∶ 2 = ( 4 − 4 )( 4 − 1) over ℚ . Its cluster picture is Here, there are two valid discs = (0, 0) and ′ = (0, 1). These correspond to the two proper clusters in the cluster picture. Using ( ) = 0 and ′ ( ) = 4, we find ) and .
Note that has a proper cluster of non-integral depth, so Theorem 7.3 does not apply verbatim; we need the more general version from Remark 7.5. We give a few illustrative charts of the model ) and  • =  ⧵ {( , , ), ( − 1, , )}. Note that the special fibre of  is non-reduced. More precisely its closure is a projective line of multiplicity 2 with self-intersection −1 and is blown down when constructing the minimal regular model (see Remark 7.4). The same applies to the component corresponding to (1,2). For the disc 1 = (1, 1), we get ) .
Note that although the reduction of 2 has a double root at 2 = 0, this double root does not correspond to a valid subdisc of 2 . Hence we do not remove this point in forming

DUAL GRAPH OF SPECIAL FIBRE AND ITS HOMOLOGY (SEMISTABLE CASE)
In this section, is semistable. Let  min be its minimal regular model over  nr . The dual graph Υ consists of a vertex Γ for every irreducible component Γ of the geometric special fibre  min , with an edge connecting Γ and Γ ′ for each intersection point of Γ and Γ ′ (self-intersections of Γ correspond to loops based at Γ ). The action of Frob on  min induces a corresponding action on Υ .
Theorem 8.1. Υ consists of one vertex for every non-übereven principal cluster and two vertices + , − for each übereven principal cluster , connected by chains of edges as follows: Here, we adopt the convention that + = − = if is not übereven, so, for example, if ′ < are even non-übereven principal clusters, then there are two chains of edges + ′ , − ′ connecting ′ and . Frobenius acts on Υ by Frob( ± ) = ± (Frob) and Frob( ) = (Frob) Frob( ) , where − denotes with the opposite orientation.
(2) If  is übereven, let be the set of those clusters ∈ such that * = .
In both cases, Frobenius acts on 1 (Υ , ℤ) is by Frob( ) = (Frob) Frob( ) , and the length pairing by where ( 6 ) is the Legendre symbol. In other words, the action on Υ is trivial if ≡ ±1 or ±5 mod 24, and interchanges the two edges + and − if ≡ ±7 or ±11 mod 24. In particular, the Frobenius action on Υ can be non-trivial even when the action on  is trivial. Pictorially, Υ is From this, we see that 1 (Υ , ℤ) = ℤ, the induced action of Frobenius is multiplication by ( 6 ), and the length pairing is ⟨ , ⟩ = 2 . This agrees with Theorem 8.2. We compute the homology 1 (Υ , ℤ) using Theorem 8.2, without first computing Υ . Except for  the even non-übereven clusters are the two twins 1 and 2 , so 1 (Υ , ℤ) is free of rank 2, generated by 1 and 2 . Frobenius acts on 1 (Υ , ℤ) by multiplication by ( −1 ), and the length pairing has matrix = ( + + ) . From this, we see that the potential toric rank of Jac is 2 (potentially totally toric reduction), and that the group of geometric components of the special fibre of the Néron model of Jac has size det( ) = + + . By computing the Smith normal form of , we find that the group structure is ℤ∕ ℤ ⊕ ℤ∕ ℤ, with = gcd( , , ) and = ( + + )∕ gcd( , , ).

SPECIAL FIBRE OF THE MINIMAL REGULAR SNC MODEL (TAME CASE)
Assume has tame reduction. We give a qualitative description of the special fibre of the minimal regular model of with strict normal crossings (SNC), over  nr . Denote this model  , special fibre ̄. We assume  is principal. † Notation 9.1.
Remark 9.4. There is also a description of the action of Gal(̄∕ ) on the special fibre in terms of clusters. Moreover, one can in principle find equations for the components of the special fibre. We refer to the references below.
Example 9.5 (A type II * elliptic curve). Take ∶ 2 = 3 − 5 over ℚ for ⩾ 5, and 3 a primitive third root of unity. † The cluster picture is  = 5∕3,  = 5,  = 6 and  = 5∕2. As ℚ ()∕ℚ is tamely ramified, has tame reduction. The cluster  is principal and fixed by , but the roots lie in a single -orbit. The special fibre of the minimal regular SNC model (displayed right) has a single central component Γ  of multiplicity 6 and genus 0, intersected by sloped chains with parameters (−1, 5∕6, 1), (−3, −5∕2, 3), and (−5∕2, −5∕3, 2) coming from the first, fourth, and fifth rows of the (second) table in Theorem 9.3 † The material in this section applies verbatim to elliptic curves of the form 2 = cubic. which are each minimal † satisfying the determinant condition of Definition 9.2. We find that the special fibre has the pictured form, so the Kodaira type of is II * .
Remark 9.6. The other Kodaira types arise similarly to the above example, with one central component met by several sloped chains.
The Γ ± are each intersected by one further chain with parameters (−2, −1∕2, 1) arising from the sixth row of the second table in Theorem 9.3.

TAMAGAWA NUMBER (SEMISTABLE CASE)
Let ∕ ∶ 2 = ( ) be a semistable hyperelliptic curve. The Tamagawa number Jac of the Jacobian of is the number of -points of the component group-scheme of the special fibre of the Néron model of Jac . We explain how to read off Jac from the cluster picture of . In general, when has übereven clusters, the formula becomes significantly more complicated, and is best phrased in the language of BY trees. For a vertex ∈ ⧵ , we write for the size of the -orbit containing . We write = ∏ −1 =0 ( ), where is the connected component of ⧵ containing . If ∈ ⧵ is an edge, then we define and similarly. We writê⊆ for the subgraph consisting of together with all vertices with = −1 and edges with = −1. Finally, we write ′ ⊆̂′ ⊆ ′ for the respective quotients of ⊆̂⊆ by the action of , with length function ′ ( ′ ) = ( ) and with ′ = for any edge ∈ mapping to ′ .

Theorem 10.3. The Tamagawa number of Jac is given by
where: (1) is the product of the sizes of all -orbits of connected components of̂; Label the edges ± , , , ± where has length and so on. Since = = 1 for all vertices and edges ,̂= , and ′ and ′ are given by the following picture, witĥ′ = ′ : There are four -orbits in̂, two of size 1 and two of size 2. Therefore, = 4. The set̂′ ⧵ ′ is empty and sõ= 1. Finally, = 3, and the set = {{ ′ , ′ , ′ }, { ′ , ′ , ′ }, { ′ , ′ , ′ }} where ′ is the image of ± and so on. Putting this together, we see Example 10.8. Consider the BY tree as in Example 10.7, but where = −1 for each component instead of 1. The edges ± and ± lie in -orbits of size 2 so ± = ± = 1, and and lie in an orbit of size 1 so = = −1. The graphs ′ and ′ are as above, and̂′ is given by There are three -orbits of components in̂, one of size 1 and two of size 2, and hence = 4. There is one connected component ′ ∈̂′ ⧵ ′ and sõ=̃( ′ ) is non-trivial. We have assumed that is even, and sõ= gcd( + , 2) = gcd( , 2) as ′ consists of two points of ′ a distance

GALOIS REPRESENTATION
In this section, we will describe the Galois action on the -adic étale cohomology of the curve (equivalently its Jacobian) when ≠ . For an arbitrary curve (or abelian variety), there always exists a decomposition of -adic Galois representations 1 ét ( ∕̄, ℚ ) = 1 ét (Jac ∕̄, ℚ ) = 1 ab ⊕ ( 1 ⊗ sp (2)  ) into 'abelian' and 'toric' parts, where for ∈ , sp(2)(Frob ) = for a choice of tame -adic character , and with = | |. We will describe the abelian and toric parts in terms of the cluster picture. Writēfor the restriction of to .
(1) For all ∈ , there exists a continuous -adic representation, , with finite image of inertia such that:

where = Stab ( ) is the Galois stabiliser of , and ⊖ is the inverse of the direct sum ⊕. (2) Let = Stab ( ) be the inertia stabiliser of and let
∶ →Q × be any character whose order is the prime-to-part of the denominator of [ ∶ ]̃. Then for all ∈ , there is an isomorphism wherẽis the set of odd children of with -action.
Remark 11.2. The full Galois module structure of cannot be determined by the cluster picture alone; indeed two curves with good reduction can have the same cluster picture but different Galois representations. It is, however, computable via a form of point-counting over finite fields; see [11, Theorem 1.5 and Example 1.9] for the statement and a worked example.
On the other hand, Theorem 11.1 (2) gives an explicit description of the inertia representation. For tame curves, it is completely determined by the underlying abstract cluster picture (in the sense of Section 17) without needing to know the inertia action on the roots a priori.
Remark 11.3. The Jacobian Jac (equivalently ) is semistable if and only if both 1 and 1 are unramified. If moreover 1 is the zero representation, then this is equivalent to Jac having good reduction. Recall from Section 5 that these conditions are easy to read off from the cluster picture of . Notation 11.4. For a cluster , we let denote the inertia stabiliser. If is coprime to , we further writeQ [ , ] to mean the -representationQ [ ∕ ] where [ ∶ ] = , and let , be a fixed faithful character of ∕ . We shall omit the cluster subscript when = . Example 11.5. Let 3 be a primitive cube root of unity and consider the curve ∕ℚ 7 ∶ 2 = (( − 7 1∕2 ) 3 − 7 5∕2 )(( + 7 1∕2 ) 3 + 7 5∕2 ), with cluster picture and  = 1 ∪ 2 ∪ {0}. In this case, inertia acts on  through a 6 -quotient and permutes 1 and 2 . We shall compute the inertia action on 1 ét ( ∕Q 7 ,Q ). Note that every cluster is odd: this implies that there is no contribution from the toric part, that is, 1 = 0, and also that ≅ ⊗ (Q [̃] ⊖ ) by definition of . Moreover, every proper cluster is principal and hence we choose representatives for ∕ ℚ 7 to be 1 and .

CONDUCTOR
In this section, we describe the conductor exponent of Jac , which we shall denote by .
For general , the formula for the conductor is more involved. Remark 12.4. If > 2g + 1, then is tame so that wild = 0 and = tame . Moreover, in this case is completely determined by the underlying abstract cluster picture (in the sense of Section 17) without needing to know the Galois action on the roots a priori.
One can check that the curve is semistable (Theorem 5.1) so we can apply Theorem 12.1. Observe that = { 1 , 2 , 3 } from which we obtain that = 2 since  is übereven.
Example 12.6. Let ∕ℚ 5 ∶ 2 = 5 + 256 and let 5 be a primitive fifth root of unity. Then the cluster picture is We begin with wild and observe that the roots form a single orbit under inertia. For all ∈ , we have that ℚ 5 ( )∕ℚ 5 has discriminant 50000, degree 5, residue degree 1 hence wild = 1.
Example 12.7. In this example, we compute the conductor directly from the cluster picture without reference to a curve. Let ⩾ 7 and let ∕ be a genus two hyperelliptic curve with = 1, with cluster picture and inertia acts by cyclically permuting the roots † in . † This is actually the only possible action due to the depths. An example of such a curve is ∕ℚ 7 ∶ 2 = 5 + 5 4

ROOT NUMBER (TAME CASE)
We give a description of the local root number ( ∕ ) of an abelian variety (for example, = Jac ), first in the case of semistable reduction, then in the case of tame reduction. For Jac , we give this description in terms of the cluster picture. Notation 13.1. Throughout, will denote a fixed character of of order , and for an abelian variety ∕ we shall decompose 1 ét ( ∕̄, ℚ ) = 1 ⊕ ( 1 ⊗ sp(2)) as in Section 11. For a cluster , let = Stab ( ) and = Stab ( ) be its Galois and inertia stabilisers, respectively, and let = [ ∶ ]. We write for the set of all cotwins and all even, non-übereven clusters of .

Theorem 13.2. Let ∕ be an abelian variety with semistable reduction. Then
When ∕ is semistable, (Jac ∕ ) may then be computed from the cluster picture as follows.  In the case of Jacobians of hyperelliptic curves with tame reduction, ⟨ , 1 ⟩ can be calculated as in Proposition 13.3 and can be read off the cluster picture.

MINIMAL DISCRIMINANT (SEMISTABLE CASE)
The discriminant Δ of is given by The following theorem provides a formula to compute the valuation of the discriminant in terms of cluster pictures.

Theorem 15.1. The valuation of the discriminant of is given by
Let (Δ min ) denote the valuation of the minimal discriminant † of the curve . If has semistable reduction, one may read off this quantity from the cluster picture or from the centred BY tree associated to the equation.
Since has semistable reduction and | | > 7, we may now apply Theorem 15.2 in order to find the valuation of the minimal discriminant. The right-hand side of the equation in that theorem is Example 15.7. Consider the curve ∶ 2 = 7( 2 + 1)( 2 + 36)( 2 + 64) defined over ℚ 7 . This is a genus 2 hyperelliptic curve with cluster picture Using one of the formulas from Theorem 15.1, we get (Δ ) = 22.
Since has semistable reduction, we can apply Theorem 15.2. Note that the two clusters 1 = { , ± 7 }, 2 = {− , − ± 7 } are permuted by Frobenius. Therefore = 1 here and the right-hand side of the formula vanishes. In particular, we find that (Δ min ) = (Δ ) = 22. The minimality of the equation is also implied by Theorem 16.3, since Condition (1) of that theorem is satisfied.
The minimal discriminant is not invariant under unramified extensions. Let denote the base change of to = ℚ 7 ( ). Since the extension is unramified, the cluster picture does not change. However, the two clusters 1  The cluster picture corresponding to this equation is . In both of the above cases, the associated BY trees consist of two blue vertices joined by a blue edge of length 2: . The centred BY trees are obtained by adding an additional vertex in the midpoint of the edge joining 1 and 2 : . From the formula in Theorem 15.5, we see that the valuation of the minimal discriminant is given by 12 + 10 ⋅ . The only difference between the (centred) BY trees corresponding to and is the action of Frobenius, and we have = 1 for and = 0 for . As before, we find (Δ min ) = 22 and (Δ min ) = 12.

References
It is minimal if the valuation of its discriminant is minimal amongst all integral Weierstrass equations. We first characterise when the equation is integral in terms of the cluster picture. Note that the cluster picture of hyperelliptic curve is unchanged by a substitution ↦ − . As a result, for a hyperelliptic curve ∕ ∶ 2 = ( ) it is not possible to check whether ( ) ∈  [ ] from the cluster picture of , but up to these shifts in the -coordinate this is possible.
for some ′ that is either empty or a -stable child ′ < with either | ′ | = 1 or ′ ⩾ 0.
We are further able to give a criterion for checking whether a given Weierstrass equation is in fact minimal. (1) There are two clusters of size g + 1 that are swapped by Frobenius,  = 0 and ( ) ∈ {0, 1}.
(2) There is no cluster of size > g + 1 with depth > 0, but there is some -stable cluster with | | ⩾ g + 1, Using examples we now illustrate how one can easily use cluster pictures and the results of this section to check whether a Weierstrass equation is integral and/or minimal.
, a genus 2 hyperelliptic curve over ℚ , for some prime > 3. Let us use the cluster picture of to test whether there exists some ∈ such that ( − ) ∈  [ ]. The cluster picture of is as follows: Note that  and are both proper and ℚ -stable, < ,  ⩽ 0, and ⩾ 0. A simple calculation gives that Therefore, by Theorem 16.1, we conclude that there exists some ∈ such that ( − ) ∈  [ ].

ISOMORPHISMS OF CURVES AND CANONICAL CLUSTER PICTURES
Definition 17.1. Let be a finite set, Σ a collection of non-empty subsets of (called clusters), and some ∈ ℚ for every ∈ Σ of size > 1, called the depth of . Then Σ (or (Σ, , )) is a cluster picture if: ∈ Σ and { } ∈ Σ for every ∈ ; two clusters are either disjoint or one is contained in the other; for , ′ ∈ Σ, if ′ ⊊ then ′ > . For a hyperelliptic curve ∕ ∶ 2 = ( ), denote the cluster picture by Σ = (Σ , , ), the collection of all clusters of  with depths.
(1) Increase the depth of all clusters by ∈ ℚ: ′ = + for all ∈ Σ.  It turns out that, provided | | > 2g + 1, every equivalence class of cluster pictures of semistable hyperelliptic curves has an 'almost canonical' representative.
Furthermore, if ′ has even genus, then we may replace (3) by the following.
In the even genus case, any other -isomorphic curve satisfying (1), (2), and (3') has the same cluster picture and valuation of leading term as .
For a semistable hyperelliptic curve ∕ , to practically use BY trees to find the canonical representative of the equivalence class of Σ , attach an open yellow edge to the centre ([10, Definition 5.13]) of . For a more detailed explanation of this, see Remarks A.8 and A.9. Example 17.6. Consider the hyperelliptic curve ∶ 2 = 6 − 1 over ℚ , for some prime ≠ 3, where Σ = . By Definition 17.2 (1), we may increase the depth of  by = 1 3 to obtain an equivalent cluster picture. Theorem 17.3 tells us there is someQ -isomorphic curve ′ ∕Q with this cluster picture. In particular, we find that under the transformations = ′ ∕ 1∕3 and = ′ ∕ , is ℚ ( 3 √ )-isomorphic to ′ ∕ℚ ( 3 √ ) ∶ ′2 = ′6 − 2 .

Consider the transformation →
This illustrates how to obtain the middle picture with = 1 5 and = 9 5 overQ 7 . All of , 1 , and 2 have the following BY tree: . Indeed, so does any other hyperelliptic curve with a cluster picture in the equivalence class of Σ . Conversely, any hyperelliptic curve ′ with BY tree ′ = would need to have its cluster picture in the equivalence class of Σ .

APPENDIX A: MINIMAL DISCRIMINANT AND BY TREES (SEMISTABLE CASE)
Throughout this section, it is assumed that is semistable. We give a proof for how to read off (Δ min ) from the BY tree associated to . Definition A.1. For a connected subgraph of a BY tree, we define a genus function by g( ) = #(connected components of the blue part) − 1 + ∑ ∈ ( ) g( ).
Note that g( ) = g as per Lemma 4.8.

Definition A.2.
If there is an edge ∈ ( ) such that both trees in ⧵ { } have equal genus (that is, genus ⌊ g 2 ⌋), then we insert a genus-0 vertex on the midpoint of , colour it the same as , and call it the centre of . Otherwise, choose ∈ ( ) such that all trees in ⧵ { } † have genus smaller than g∕2. In both cases, the centred BY tree * is the tree with vertex set ( * ) = ( ) ∪ { }; we denote by ⪯ the partial order on ( * ) with maximal element . For a vertex ∈ ( * ), we say that the vertex connected to lying on the path to the centre of * is its parent. All other vertices connected to are called children of . The centre itself does not have a parent. For a connected subgraph of , we set ( ) = ∑ ∈ ( ).
• There is a unique vertex ∈ ( ) with the property that ( ) < g + 1 for all trees in ⧵ { }. • There is a unique edge ∈ ( ) with the property that ( ) = g + 1 for both trees in ⧵ { }.
Further, g( ) = ⌊ ( )−1 2 ⌋ for any connected subgraph of a BY tree (see [10,Remark 5.14]). This shows that the centre of a BY tree is indeed well defined.  Proof. Let Σ = Σ be the cluster picture associated to , see Definition 17.1 for the definition of abstract cluster pictures. We associate a cluster picture Σ 1 = (Σ 1 , 1 , 1 ) to the centred tree * in the following way. For every vertex ∈ * , define where { , } are singletons. For ≠ , the relative depth of the cluster is given by = . We have = 1 and assign to it depth 1 = 0. The construction of the cluster picture coincides with Construction 4.15 in [10], although phrased in a slightly different language (cf. Remark A.9). Therefore, the BY tree associated to this cluster picture is . Moreover, it is clear from the construction that for every vertex ∈ ( * ), we have ( ) = | | and that every cluster ≠  has size ⩽ g + 1.
From Theorems 17.3 and 17.4, it follows that there is a hyperelliptic curve 1 ∶ 2 = 1 ( ) which is̄-isomorphic to and has cluster picture Σ 1 . Applying the formula of Theorem 15.1, we find that where 1 denotes the leading coefficient of 1 . We will now modify the cluster picture Σ 1 in order to find a curve 2 which is isomorphic to over . Let us first consider the case where ∈ ( ). In that case, we moreover have that | | < g + 1 for all clusters ≠ . It follows from Theorem 17.5 and the uniqueness of the centre that there is a -isomorphic curve 2 ∶ 2 = 2 ( ) with cluster picture Σ 2 = Σ 1 and ( 2 ) = 0, where 2 is the leading coefficient of 2 . This completes the first case. Now consider the case ∉ ( ). Then  = 1 ⊔ 2 , where | 1 | = | 2 | = g + 1. In this case, it might be necessary to redistribute depth between the clusters 1 and 2 , see Definition 17.2. However, this does not change the valuation of the discriminant since the two clusters have equal size. Hence, we may still use equation (A.1). If the two clusters 1 and 2 are not permuted by Frobenius, let Σ 2 be the cluster picture obtained by redistributing all depth from 1 to 2 (or vice