The ℓ-parity conjecture over the constant quadratic extension

Česnavičius, Kęstutis

doi:10.1017/s030500411700055x

Cited by 6 publications

(11 citation statements)

References 33 publications

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“…In this appendix, for a principally polarised abelian variety A over a number field K, we study the parity of dim F 2 X 0 (A/L) [2] where L/K is a finite field extension. The result is Proposition A.1 and has been independently observed by Česnavičius [6,Lemma 3.4]. As a consequence, we remove the assumption in two theorems of T. and V. Dokchtiser ([12, Theorem 1.6(b)] and [13,Theorem 1.6]) that the princpal polarisation on the abelian variety in question is induced by a rational divisor.…”

mentioning

confidence: 75%

“…Theorem 1.3 (or rather Theorem 3.2 from which we deduce it) is likely known to experts though we have not found it in the literature in this generality. Klagsbrun, Mazur and Rubin give an alternate proof of the elliptic curves case, originally due to Kramer, in [19,Theorem 3.9] and Česnavičius generalises their setup to higher dimension in [6,Theorem 5.9]. This would likely give an alternate approach to proving Theorem 1.3.…”

Section: Introductionmentioning

confidence: 99%

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2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions

Morgan¹

2015

Preprint

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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 it over quadratic extensions of the base field, providing essentially the first examples of the 2-parity conjecture in dimension greater than one. 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. Particularly surprising is the appearance in the formula of terms that govern whether or not the Cassels-Tate pairing on the Jacobian is alternating, which first appeared in a paper of Poonen and Stoll. We prove this local formula in many instances and show that in all cases it follows from standard global conjectures.

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confidence: 75%

Section: Introductionmentioning

confidence: 99%

2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions

Morgan¹

2015

Preprint

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“…Proof Let

scriptH

be the Cartier dual of

scriptG

. By [, 5.3], the images of

{prefixloc}^{n} false(scriptG false)

and

{prefixloc}^{2 - n} false(scriptH false)

are orthogonal complements under the sum of cup product pairings

H^{n} false(K_{v}, scriptG false) \times H^{2 - n} false(K_{v}, scriptH false) \to H^{2} false(K_{v}, G_{m} false) \overset{{inv}_{v}}{\to} Q / Z .

By Proposition and the discreteness of

H^{2} false(K_{v}, G_{m} false)

supplied by [, 3.5 (b)], these pairings are continuous, so the claim follows.

□

…”

Section: Discreteness Of the Image Of Global Cohomologymentioning

confidence: 99%

“…Proof Exactness up to

H^{1} false(U, scriptG false)

amounts to and the definition of

Sel (G)

. By [, 5.3],

Im false({loc}^{1} (G) false) \subset ⨁_{v \notin U} H^{1} false(K_{v}, scriptG false) and Im false({loc}^{1} (H) false) \subset ⨁_{v \notin U} H^{1} false(K_{v}, scriptH false)

are (closed) orthogonal complements under the sum of the pairings , and hence, by [, III.28, Corollary 1] and [, 24.10], so are

Im false({loc}^{1} (G) false) + ⨁_{v \notin U} Sel false(G_{K_{v}} false) \subset ⨁_{v \notin U} H^{1} false(K_{v}, scriptG false) and Im false({loc}^{1} (H) |_{prefixSel false(scriptH false)} false) \subset ⨁_{v \notin U} H^{1} false(K_{v}, scriptH false) .

We therefore arrive at further orthogonal complements

Im (H^{1} (U, G) \to ⨁_{v \notin U} H^{1} (K_{v}, G)<...>)

…”

Section: Cassels–poitou–tatementioning

confidence: 99%

p-Selmer growth in extensions of degree p

Česnavičius

2017

J. London Math. Soc.

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Abstract. There is a known analogy between growth questions for class groups and for Selmer groups. If p is a prime, then the p-torsion of the ideal class group grows unboundedly in Z{pZ-extensions of a fixed number field K, so one expects the same for the p-Selmer group of a nonzero abelian variety over K. This Selmer group analogue is known in special cases and we prove it in general, along with a version for arbitrary global fields.1. Introduction 1.1. Growth of class groups and of Selmer groups. It is a classical theorem of Gauss that the 2-torsion subgroup PicpO L qr2s of the ideal class group of a quadratic number field L can be arbitrarily large. Although unboundedness of # PicpO L qrps for an odd prime p is a seemingly inaccessible conjecture, [BCH`66, VII-12, Thm. 4] explains how to extend Gauss' methods to prove that # PicpO L qrps is unbounded if L{Q ranges over the Z{pZ-extensions instead.As explained in [Čes15a], growth questions for ideal class groups and for Selmer groups of abelian varieties are often analogous. It is therefore natural to hope that for a prime p and a nonzero abelian variety A over Q, the p-Selmer group Sel p A L can be arbitrarily large when L{Q ranges over the Z{pZ-extensions. Our main result confirms this expectation. Theorem 1.2 (Theorem 5.6). Let p be a prime, K a global field, and A a nonzero abelian variety overis unbounded when L ranges over the Z{pZ-extensions of K. Remarks.1.3. In the excluded case when ArpspKq " 0 and A is not supersingular (when also char K " p and dim A ą 2), there nevertheless is an n P Z ą0 that depends on A such thatis unbounded when L ranges over the Z{p n Z-extensions of K, see Theorem 5.6.1.4. See Corollary 5.5 for a version of Theorem 1.2 for Selmer groups of arbitrary isogenies.1.5. The proof of Theorem 1.2 also reproves the unbounded growth of the p-torsion subgroup of the ideal class group in Z{pZ-extensions of K, see Corollary 5.3.

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“…Then the function ǫ is not a homomorphism for J/Q so that half of the 2-Selmer ranks of the quadratic twists of J are even and half odd. On the other hand, J has κ = 3 16 and odd 2 ∞ -Selmer rank, so that 19/32 of the twists of J have even 2 ∞ -Selmer rank and 13/32 odd.…”

Section: Introductionmentioning

confidence: 99%

Quadratic twists of abelian varieties and disparity in Selmer ranks

Morgan

2019

Alg. Number Th.

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We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun-Mazur-Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich-Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree and we describe both.

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The ℓ-parity conjecture over the constant quadratic extension

Cited by 6 publications

References 33 publications

2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions

2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions

p-Selmer growth in extensions of degree p

Quadratic twists of abelian varieties and disparity in Selmer ranks

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