Indices of hyperelliptic curves over p -adic fields

Geel, Jan Van; Yanchevskii, V. I.

doi:10.1007/s002290050070

Cited by 4 publications

(9 citation statements)

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“…The papers [vGY1] and [vGY2] give criteria in special cases for when a hyperelliptic curve of genus g over a p-adic field admits a line bundle of odd degree. The first half of the following lemma corresponds to Proposition 3.1 in [vGY1].…”

Section: The Criterion For Oddness Of Jacobiansmentioning

confidence: 99%

The Cassels-Tate Pairing on Polarized Abelian Varieties

Poonen¹,

Stoll²

1999

The Annals of Mathematics

147

210

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Let (A, λ) be a principally polarized abelian variety defined over a global field k, and let ∐ ∐(A) be its Shafarevich-Tate group. Let ∐ ∐(A) nd denote the quotient of ∐ ∐(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairingIf A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric.Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on ∐ ∐(A) nd . These criteria are expressed in terms of an element c ∈ ∐ ∐(A) nd that is canonically associated to the polarization λ. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether #∐ ∐(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g ≥ 2 over Q have a Jacobian with nonsquare #∐ ∐ (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.

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Section: The Criterion For Oddness Of Jacobiansmentioning

confidence: 99%

The Cassels-Tate Pairing on Polarized Abelian Varieties

Poonen¹,

Stoll²

1999

The Annals of Mathematics

147

210

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“…We caution that [20] contains some minor errors. These are discussed and corrected in the PhD thesis of Nowell [45] (see also [2, Section 9]). Remark For an alternative, but related, approach to constructing regular models of hyperelliptic curves over non‐archimedean local fields of odd residue characteristic, see the works of Srinivasan [59] and Obus–Srinivasan [46].…”

Section: Ramified Quadratic Twists Of Semistable Hyperelliptic Curvesmentioning

confidence: 99%

“…As mentioned, Proposition 14.5 will follow from results of Faraggi and Nowell, specifically from [20, Theorems 7.12 and 7.18] and [45, Theorems 9.23, 9.31 and 9.32]. These results take as input several invariants of hyperelliptic curves.…”

Section: Ramified Quadratic Twists Of Semistable Hyperelliptic Curvesmentioning

confidence: 99%

2‐Selmer parity for hyperelliptic curves in quadratic extensions

Morgan

2023

Proceedings of London Math Soc

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

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“…Let The cases 03B103C0 mod k*2 and 03B103C0 mod k*2, A2 = 03B103C0 mod k*2 reduce to case (2) and (4) (The choice 03C0' = -03C0 avoids making a distinction between fields k in which -1 is a square and those in which -1 = a mod k*2.) In the first case the splitting fields of fi and f2 are both equal to k( f) and in the last case they are both equal to k( Proposition 3.8 in [5] then implies that the index of is 2 in both these cases. Propositions 3.1, 3.2, 3.4 and 3.6 determine the index in the remaining four cases.…”

Section: Introductionmentioning

confidence: 98%

“…In [5] the authors investigated equations of the form with h(X ) = hiXi a polynomial without multiple roots over such a field k. It is well known that such equations define an affine plane curve C( which is isomorphic with the affine part of a smooth geometrically connected projective curve Ch over k (cf. [6]).…”

Section: Introductionmentioning

confidence: 99%