Quadratic twists of abelian varieties and disparity in Selmer ranks

Morgan, Adam

doi:10.2140/ant.2019.13.839

Cited by 6 publications

(17 citation statements)

References 25 publications

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“…= −1, which shows that J/G is the quadratic twist of J by K. Proof. By [30,Lem. 4.16], the quadratic twist of a principally polarized abelian variety is itself principally polarized.…”

Section: Ranks Of Jacobians Of Trigonal Curvesmentioning

confidence: 99%

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3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Bhargava¹,

Klagsbrun²,

Oliver³

et al. 2019

Duke Math. J.

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For an abelian variety A over a number field F , we prove that the average rank of the quadratic twists of A is bounded, under the assumption that the multiplication-by-3 isogeny on A factors as a composition of 3-isogenies over F . This is the first such boundedness result for an absolutely simple abelian variety A of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field.In dimension one, we deduce that if E/F is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If F is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have 3-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld's conjecture -which states that the average rank in the quadratic twist family of an elliptic curve over Q should be 1/2 -and the first progress towards the analogous conjecture over number fields other than Q.Our results follow from a computation of the average size of the φ-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny φ.

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“…= −1, which shows that J/G is the quadratic twist of J by K. Proof. By [30,Lem. 4.16], the quadratic twist of a principally polarized abelian variety is itself principally polarized.…”

Section: Ranks Of Jacobians Of Trigonal Curvesmentioning

confidence: 99%

“…Proof. By [30,Lem. 4.16], the quadratic twist of a principally polarized abelian variety is itself principally polarized.…”

Section: Ranks Of Jacobians Of Trigonal Curvesmentioning

confidence: 99%

3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Bhargava¹,

Klagsbrun²,

Oliver³

et al. 2019

Duke Math. J.

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“…Now take φ as in the theorem statement, and choose an even positive integer n divisible by the order of φ. Denote by (1, λ) : A → A × A ∨ the composition of 1 × λ with the diagonal embedding of A into A × A, and denote by P the Poincaré line bundle on A × A ∨ . Then L 1 = (1, λ) * P n 2 /2 is a symmetric line bundle on A, and arguing as in [32,Lemma 4.13] we see that the base-change of L 1 to F s is isomorphic to L n 2 . Take H to be the theta group associated to L 1 , so we have an exact sequence…”

Section: Proposition 73 With the Notation Above Writementioning

confidence: 52%

“…Equivalently, one sees from [38, Corollary 2.2] that ψ PS is the class in H 1 (G F , T /2T ) corresponding to the G F -set of quadratic forms on T /2T refining the pairing P 0 (cf. [32,Section 3A]). This directly extends Poonen and Stoll's work to e.g.…”

Section: Proposition 73 With the Notation Above Writementioning

confidence: 99%

“…Nevertheless, it is possible to use the work of Poonen-Stoll [40] to analyze the failure of this pairing to be alternating, allowing one to once again decompose the parity of the 2-infinity Selmer rank of A over K as a sum of local terms. This is carried out in work of the first author [32,Theorem 10.20], where the relative simplicity of (9.17) compared to either E or E χ plays an essential role.…”

Section: Then We Have An Exact Sequence Ofmentioning

confidence: 99%

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The Cassels-Tate pairing for finite Galois modules

Morgan¹,

Smith²

2021

Preprint

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Given a global field F with absolute Galois group G F , we define a category SMod F whose objects are finite G F -modules decorated with local conditions. We define this category so that 'taking the Selmer group' defines a functor Sel from SMod F to Ab. After defining a duality functor ∨ on SMod F , we show that every short exact sequencewhose left and right kernels are the images of Sel M and Sel M ∨ , respectively. This construction generalizes the Cassels-Tate pairing defined on Shafarevich-Tate groups of abelian varieties over global fields. Our main results generalize the fact that the classical pairing is antisymmetric for principally polarized abelian varieties but, by work of Poonen and Stoll, not necessarily alternating.We can express many previously-defined class field theoretic pairings as instances of this construction, giving the study of these pairings access to our general results for SMod F . As one application, we reprove and extend recent results about the class groups of number fields containing many roots of unity.

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Arithmetic statistics of Prym surfaces

Laga

2022

Math. Ann.

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We consider a family of abelian surfaces over $$\mathbb {Q}$$ Q arising as Prym varieties of double covers of genus-1 curves by genus-3 curves. These abelian surfaces carry a polarization of type (1, 2) and we show that the average size of the Selmer group of this polarization equals 3. Moreover we show that the average size of the 2-Selmer group of the abelian surfaces in the same family is bounded above by 5. This implies an upper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonen and Rains for a family of abelian varieties which are not principally polarized. The proof is a combination of an analysis of the Lie algebra embedding $$F_4\subset E_6$$ F 4 ⊂ E 6 , invariant theory, a classical geometric construction due to Pantazis, a study of Néron component groups of Prym surfaces and Bhargava’s orbit-counting techniques.

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Quadratic twists of abelian varieties and disparity in Selmer ranks

Cited by 6 publications

References 25 publications

3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

The Cassels-Tate pairing for finite Galois modules

Arithmetic statistics of Prym surfaces

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