Differential Forms on Hyperelliptic Curves with Semistable Reduction

Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$ its minimal regular model with relative dualizing sheaf $\omega_{\mathcal{X}/ \mathcal{O}_K}$. We show how to directly read off a basis for $H^0(\mathcal{X},\omega_{\mathcal{X}/\mathcal{O}_K})$ from the cluster picture of the roots of $f$. Furthermore we give a formula for the valuation of $\lambda$ such that $\lambda \cdot \frac{dx}{y} \land \dots \land x^{g-1}\frac{dx}{y}$ is a generator for $\det H^0(\mathcal{X},\omega_{\mathcal{X}/\mathcal{O}_K})$.


Introduction
Let (R, v) be a discrete valuation ring with maximal ideal p = (π) and field of fractions K. For an element r ∈ R, we writer for the reduction modulo p. The separable closure of K is denoted by K sep and the algebraic closure byK. We assume that the residue field k = R p has characteristic p ≠ 2.
Let C K be a hyperelliptic curve given by a Weierstraß equation Throughout this paper, we will always assume that the curve C has semistable reduction and genus g > 1. Since p ≠ 2, it is easy to read off from the polynomial f whether C is semistable and determine the semistable reduction. This is described in [1] and [2]. To a Weierstraß equation, we associate the differential forms These form a basis of H 0 (C, Ω C K ). Let X → SpecR be the minimal regular model of C and ω X R the relative dualizing sheaf [5,Definition 6.4.18.]. In our situation, the relative dualizing sheaf is isomorphic to the canonical sheaf [5,Theorem 6.4.32]. We have that H 0 (C, Ω C K ) = H 0 (X , ω X R ) ⊗ R K and H 0 (X , ω X R ) is a free R-module of rank g. So det H 0 (X , ω X R ) ∶= ⋀ g H 0 (X , ω X R ) is free of rank one over R.
2. Explicitly determine a basis for the global sections of ω X R .
Our approach is based on results of [4]. Under simplified hypotheses a formula for λ C and a description of a basis for the global sections of ω X R is given in Proposition 5.5. of that paper.

Motivation
Our motivation for studying the differential forms on the minimal regular model comes from the Birch and Swinnerton-Dyer conjectures. Originally formulated for elliptic curves, the conjectures were later generalized to abelian varieties over number fields by Tate [6].
Let C be a hyperelliptic curve defined over the rational numbers Q and let J denote its Jacobian. The second BSD conjecture in this situation is Here L(J, s) is the L-series of J and r its analytic rank. Reg denotes the regulator of J(Q). For a prime p, the Tamagawa number is denoted by c p . The Shafarevich-Tate group is represented by X(J, Q) and J(Q) tors is the torsion subgroup of J(Q). The results of the present paper can be applied to calculate the sixth quantity, that is the period Ω. For the description of this quantity we follow the outline in [7,Section 3] and [3,Section 3.5]. Both papers provide numerical evidence for the BSD conjectures for hyperelliptic curves.
Let J → SpecZ denote the Néron model of J. We have that H 0 (J , Ω J Z ) is a free Zmodule of rank g. Let (µ 0 , . . . , µ g−1 ) be a basis for this module. Then µ ∶= µ 0 ∧ ⋅ ⋅ ⋅ ∧ µ g−1 is a generator for ⋀ g H 0 (J , Ω J Z ) and Ω is defined as Finding a basis for H 0 (J , Ω J Z ) can be done locally. Let p ∈ Z be a prime. We write R = Z p and use the notation introduced in the beginning. We have that Ω J R is isomorphic to the canonical sheaf ω X R , where X is the minimal regular model of C. So it is enough to find a basis for the global sections of ω X R .
For computational purposes one does not necessarily need to know the basis of regular differentials. In [7] and [3] the authors first evaluate ∫ J(R) ω , where ω is the exterior product of the differentials ω 0 , . . . , ω g−1 associated to the Weierstraß equation for C and then compute a correction term in order to get the value Ω ∶= ∫ J(R) µ .

Results
We give a brief overview of our main results and illustrate them by means of an example. The results are stated in terms of cluster pictures. A cluster picture is a combinatorial object associated to the equation of a curve. It encodes different invariants of the curve. The concept has been studied in [2]. For definitions we refer to [2] or Section 2 of the present paper. Here we only recall the necessary notation.
The following theorem shows how to read off a basis for H 0 (X , ω X R ) from the cluster picture of a curve. Theorem 1.1 (Theorem 4.1). Let C K be a hyperelliptic curve defined by an integral Weierstraß equation C ∶ y 2 = f (x) and Σ the associated cluster picture.
Let X R be the minimal regular model. Assume that the residue field k is algebraically closed. Then an R-basis for the global sections of the relative dualizing sheaf ω X R is given by (µ 0 , . . . µ g−1 ), where The clusters s 0 , . . . , s g−1 are chosen inductively such that If the maximal value is obtained by two different clusters s and s ′ such that s ′ ⊂ s, we choose s i = s.
We are going to illustrate the procedure described in this theorem by an example.
Example 1.2. Let p > 3 and C Q p the hyperelliptic curve of genus g = 5 defined by The proper clusters are These clusters have depths d R = 0, d t 1 = 4, d t 2 = 6, d t 3 = 8 and relative depths δ t 1 = 4, δ t 2 = 2, δ t 3 = 8. This information is contained in the cluster picture Σ: The subscript of the top cluster is its depth. The subscripts of the other clusters are their relative depths. We construct a basis for H 0 (X , ω X R ) as described in Theorem 1.1. First we choose s 0 to be the cluster that maximises νt 2 − d t . The evaluation of this term for each cluster can be found in the first column (after the double line) of the table below. Next we choose s 1 to be the cluster that maximises νt 2 − d t − d t∧s 1 , the cluster s 2 is the one that maximises νt 2 − d t − d t∧s 0 − d t∧s 1 and so on.
In each column the maximal value is circled to indicate which cluster is chosen in the respective step. Three dots always represent the entire expression in the previous column. We can read off from the table and e 0 = 9, e 1 = 8, e 2 = 4, e 3 = 0, e 4 = 0.
If we choose z R = z t 1 = z t 2 = 0 and z t 3 = 1 as centres for the clusters, we get the following basis for the global sections of ω X R : As mentioned before, it is often not necessary to determine the basis for H 0 (X , ω X R ) explicitly, but it suffices to know a generator for det H 0 (X , ω X R ). Theorem 1.3 gives a convenient formula for the determination of this generator.
Let us revisit the above example.
. Note that the value v(λ C ) = 21 is equal to the sum over the e i determined by Theorem 1.1.

Outline
In Section 2, we review some definitions and facts about cluster pictures. In Section 3 we translate Proposition 5.5. of [4] into the language of cluster pictures and generalise it in order to prove Theorem 1.3. The last section is dedicated to the proof of Theorem 1.1.
Acknowledgements I would like to thank Vladimir Dokchitser for proposing to work on this topic and his support throughout the creation of this paper. I would also like to thank Stefan Wewers for very helpful discussions and comments on earlier versions of this paper, as well as Adam Morgan who also suggested a proof of Proposition 3.3.

Cluster Pictures
In this section, we describe the cluster picture associated to an equation defining a hyperelliptic curve and briefly introduce the notation used in the subsequent sections. All information is taken from [2]. Let C K be a hyperelliptic curve defined by a Weierstraß equation C ∶ y 2 = f (x). We write R for the set of roots of f (x) in K sep and c f for its leading coefficient, so that We say that z is a center of the cluster and write z = z s .
(ii) ([2] Definition 1.1.) If s > 1, then s is called a proper cluster and its depth is defined to be d s = min r,r ′ ∈s v(r − r ′ ).
(iii) ([2] Definition 1.4.) A cluster s is principal if s ≥ 3, except if either s = R is even and has exactly two children, or if s has a child of size 2g.
(iv) ([2] Definition 1.3.) If s ′ ⊊ s is a maximal subcluster, we write s ′ < s. For two clusters (or roots) s, s ′ we write s ∧ s ′ for the smallest cluster that contains them.
(v) ([2] Definition 1.5.) If s ≠ R, the relative depth of s is defined as the difference between the depth of s and the depth of the smallest cluster strictly containing s. It is denoted by δ s .
See Example 1.2 in the Introduction for an illustration of these definitions.
Remark 2.2. Note that for a root r ∈ R and a cluster s, we have So ν s can also be calculated via There is a notion of equivalence for cluster pictures that respects isomorphisms of hyperelliptic curves. For a complete discussion of the topic we refer to [2, Section 14]. The following proposition is important for the proof of Theorem 3.1. So we state it here for the convenience of the reader. ). Let f (x) ∈ K[x] be a separable polynomial with roots R ⊂K, such that G K acts tamely on R, and let Σ be the associated cluster picture. Suppose Σ ′ is a cluster picture obtained from Σ by one of the following constructions: 1. Increasing the depth of all clusters by some n ∈ Z; 2. Adding a root to R, provided R is odd, d R ∈ Z and k > #{s < R ∶ s is G K -stable}; 3. Redistributing the depth between s and R s by decreasing the depth of s by 1, is a hyperelliptic curve, then there is a K-isomorphic curve given by a Weierstraß model whose cluster picture is Σ ′ .
3 A Basis for det H 0 (X , ω X R ) Let C K be a semistable hyperelliptic curve of genus g defined by C ∶ y 2 = f (x) with f (x) = c f ∏ r∈R (x−r). We write ω 0 = dx y , . . . , ω g−1 = x g−1 dx y for the differentials associated to this equation and ω ∶= ω 0 ∧ ⋅ ⋅ ⋅ ∧ ω g−1 ∈ det H 0 (C, Ω 1 C K ). Let X → SpecR be the minimal regular model of C. The main result of this section is Theorem 3.1, where we determine λ C ∈ K, such that λ C ⋅ ω generates det H 0 (X , ω X R ) as an R-module. Note that λ C is only well-defined up to a unit. Moreover it is not a curve invariant, but depends on the equation.
Theorem 3.1. Let C K be a semistable hyperelliptic curve of genus g defined by y for the differentials associated to this equation and ω ∶= ω 0 ∧ ⋅ ⋅ ⋅ ∧ ω g−1 ∈ det H 0 (C, Ω 1 C K ). Let X → SpecR be the minimal regular model of C. Suppose that λ C ⋅ ω is a basis for det H 0 (X , ω X R ). Then A result in [4] shows that this formula is true under some additional assumptions (Lemma 3.2). We show how one can reduce to this case and work out the necessary correction terms.
Then v(λ C ) can be computed using Equation 1.
Proof. Under the conditions in the lemma, Equation 1 reduces to Note that the conditions in the above Lemma imply semistability, see for example [2, Thoerem 7.1.]. Conversely, after a tamely ramified extension of the base field, there always exists an equation for C satisfying the conditions listed in the above Lemma if C is semistable. Proposition 3.3. Let C K be a semistable hyperelliptic curve with minimal regular model X → SpecR. Let K ′ K be a finite field extension. Write C ′ for the base-change of C to K ′ , R ′ = O K ′ and X ′ → SpecR ′ for the minimal regular model of C ′ . Then Proof. Let Y → SpecR and Y ′ → SpecR ′ be the stable models of C K and C ′ K ′ respectively. The stable model Y is obtained from X by contraction of all components Γ of the special fibre for which K X R .Γ = 0. Write f ∶ X → Y for the contraction morphism. Since the intersection matrix of the contracted components is negative definite, it follows from [5,Corollary 9.4.18 We have Y ′ = Y × R R ′ and by [5,Theorem 6.4 Lemma 3.4. Let C K be a hyperelliptic curve with semistable reduction defined by C ∶ y 2 = f (x). Let K ′ K be a tamely ramified extension and write C ′ for the basechange of C to K ′ . Then Equation 1 holds for C K if and only if it holds for C ′ K ′ .
Proof. Let e K ′ K denote the ramification degree of the extension K ′ K. We write Σ for the cluster picture associated to C ∶ y 2 = f (x) and Σ ′ for the cluster picture associated to the equation C ′ ∶ y 2 = f (x) over K ′ . The clusters themselves do not change under a ramified extension, but their depths do. More precisely we have δ ′ s = e K ′ K ⋅ δ s for each cluster s and d ′ R = e K ′ K ⋅ d R . We write R ′ ∶= O K ′ and v ′ for the normalized valuation. That is v ′ (r) = e K ′ K ⋅ v(r) for all r ∈ R.

From Proposition 3.3 it follows that
. So the above calculation shows that Formula 1 is true for C K if and only if it is true for C K ′ .
Definition 3.5. Let C K be a hyperelliptic curve of genus g, defined by some Weierstraß equation y 2 = f (x). We denote by c f the leading coefficient of f . Then the discriminant ∆ of the equation is defined as While the discriminant defined above is not a curve invariant but depends on the equation, there exists a more natural definition of discriminant. See also the paragraph before Proposition 2.2. in [4]. Definition 3.6. Let C K be a hyperelliptic curve of genus g, defined by some Weierstraß equation y 2 = f (x) with discriminant ∆. We associate to this equation the differential forms ω 0 , . . . , ω g−1 and write ω = ω 0 ∧ ⋅ ⋅ ⋅ ∧ ω g−1 ∈ H 0 (C, Ω 1 C K ). Then the element is called hyperelliptic discriminant of C.
The following proposition shows that Λ is well defined.
Proposition 3.7. Let C K be a hyperelliptic curve with hyperelliptic discriminant Λ.
Let y 2 = f (x) be some Weierstraß equation for C. We associate to this equation the elements ∆ and λ. Then the following statements are true.
1. The element Λ is independent of the choice of equation.
2. Viewed as a rational section of (det H 0 (X , ω X R )) ⊗8g+4 , the order of vanishing in p is given by 3. Let y ′2 = g(x ′ ) be another equation defining the same curve with ∆ ′ and λ C ′ the corresponding quantities. Then
3. This is a direct consequence of the first two statements. Proof.
Since v(c f ) is even, c f is a square in K nr and the two equations define isomorphic curves over K nr . By definition, the discriminant of the equation Using part 3 of Proposition 3.7, we get Lemma 3.9. Let C K be a hyperelliptic curve and y 2 = f (x) a Weierstraß equation defining this curve. Let Σ be its cluster picture and λ C the quantity associated to this equation.
Let y ′2 = g(x ′ ) be a different equation for C. Denote by Σ ′ and λ C ′ the corresponding elements.
1. If Σ ′ is obtained from Σ by increasing the depths of all clusters by some t ∈ Z, then 3. If Σ has even size and Σ ′ is obtained from Σ by redistributing the depth between s < R and R s to Proof. From [2, Lemma 16.6.], we know how the discriminant changes under the above modifications of the cluster picture. We will combine these results with Proposition 3.7, part 3.
Proof of Theorem 3.1. Let C K be a hyperelliptic curve with semistable reduction defined by In Lemma 3.4 we have seen that it suffices to prove that the formula holds after a tamely ramified extension. So we may assume that R ⊂ R, v(r − s) ∈ 2Z for all r, s ∈ R and v(c f ) ∈ 2Z. Subtracting the correction term 4 g ⋅ v(c f ) from the right hand side, we may even assume that v(c f ) = 0. This follows from Lemma 3.8. Further we can perform a Möbius transformation such that the cluster picture corresponding to the new equation has outer depth d R = 0 (see Proposition 2.3, part 1). By Lemma 3.9 this corresponds to subtracting . Note that decreasing the absolute depths does not change any relative depths.
In case that R = 2g + 1, we can perform a Möbius transformation that corresponds to adding one root to the cluster picture (as in Proposition 2.3, part 2). Since d R = 0, this does not change the valuation of λ C . After these two steps we are left with proving the simplified formula that already appeared in Lemma 3.2. That is with Conditions (i)-(iv) of the lemma being satisfied and d R = 0. Without loss of generality we may always assume G K -stability for clusters because the formula for λ C behaves well under tamely ramified extension (see Lemma 3.4). So we can apply Part 4 of Proposition 2.3 that is redistribute depth between a (G K -stable) cluster s and R s. With this method we can manipulate the cluster picture such that R has at least three maximal subclusters. Together with the fact that d R = 0 this implies Condition (v) of Lemma 3.2. Assuming that the formula behaves well when redistributing depths, this proves the theorem. So we only have to show the latter.
Let s * < R be a cluster in Σ. Let Σ ′ be the cluster picture obtained after redistributing depth between s * and R s * . That is d ′ s * = d s * − t and d ′ R s * = d R s * + t for some t ∈ Z. We have already seen in Lemma 3.9, part 3 that The calculation below shows that this equals the change on the right hand side of the equation.
s even s≠R Let C K be a semistable hyperelliptic curve defined by a Weierstraß equation . To this equation we associate the cluster picture Σ. Let X → SpecR be the minimal regular model of C.
In this section we show how to read off a basis for the global sections of the canonical sheaf ω X R from the cluster picture Σ.
Theorem 4.1. Let C K be a hyperelliptic curve defined by an integral Weierstraß equation C ∶ y 2 = f (x) and Σ the associated cluster picture.
Let X R be the minimal regular model. Assume that the residue field k is algebraically closed. Then an R-basis for the global sections of the relative dualizing sheaf ω X R is given by (µ 0 , . . . µ g−1 ), where Then for any principal cluster s ∈ Σ, we have that − νs 2 − ∑ i−1 j=0 d s j ∧s − d s is a lower bound for the order of the element ∏ i−1 j=0 (x − z s j ) dx y on the component of the special fibre of X that corresponds to s.
Proof. Let s be a principal cluster in Σ and denote by Γ s the component (or possibly the two components) of the special fibre of X R corresponding to this cluster. The connection between the cluster picture of a curve and the special fibre of its minimal regular model is explained in [2,Theorem 8.5.].
The non-singular points on Γ s are all visible on the chart U P, where P is a finite set of points and U ∶= Spec R[x s , y s ] (y 2 s − f s (x s )) with local coordinates x s = x − z s π ds , y s = y π νs 2 . This follows from [2,Proposition 5.5.]. Because the set P has codimension 2 in X , it suffices to prove the statement for U P (see for example [5,Theorem 4.1.14.]).
We can write Since d s ≥ d s∧s j and zs−zs j π d s∧s j ∈ R for all clusters s, s j ∈ Σ, the element ∏ j=0 (x − z s j ) dx y is non-negative on every such component. For the horizontal part, we have to consider the restriction of µ i to the generic fibre. Clearly µ i ⊗ R K ∈ H 0 (X , ω X R ) ⊗ R K = H 0 (C, Ω C K ). This proves the first claim.
Claim 2: Let λ C be the quantity defined in the previous section, then ∑ g−1 i=0 e i = v(λ C ).
In the last line we used that ∑ g−1 i=0 #{s j ⊆ s j ≤ i} = γ(s)(γ(s)+1) 2 for every cluster s. The value of γ(s) is given in Lemma 4.2. Now the claim follows from Theorem 3.1.
So by Theorem 3.1, µ is a basis for det H 0 (X , ω X R ).