The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

Bhargava, Manjul; Elkies, Noam D.; Ari, Shnidman,

doi:10.1112/jlms.12271

Cited by 20 publications

(36 citation statements)

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“…For any place p of F , we write F p for the p-adic completion of F . From the results of [1], one naturally expects that the average number of non-trivial elements of Sel φ (A s ) is governed by an Euler product whose factor at p is the average size of the local φ-Selmer ratio…”

Section: Resultsmentioning

confidence: 99%

“…We also provide the first known examples of absolutely simple abelian varieties A over Q having dimension greater than one for which the average rank of A s (Q) is bounded, and indeed we give such examples over any number field F and in arbitrarily large dimension. 1 The proofs of these results are accomplished through a computation of the average size of a family of Selmer groups, namely, the φ-Selmer groups associated to a quadratic twist family of 3-isogenies φ s : A s → A ′ s of abelian varieties. There have been a number of recent results on the average sizes of Selmer groups in large universal families of elliptic curves or Jacobians of higher genus curves (see, e.g., [2,3,5,6,39,40,44]) obtained via geometry-of-numbers methods.…”

Section: Introductionmentioning

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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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

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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“…VIII, Exercise 8.1 with m = 2. In particular, c(E) is bounded by a multiple of the number of places of bad reduction of E plus the 2-rank of the class group of K = Q(E[2]) = Q(E (D) [2]).…”

Section: Since the Faltings Height Satisfiesmentioning

confidence: 99%

“…The proof of Theorem 1.6 combines Theorem 1.1 with results of Fouvry [12] on upper bounds for the average size of 3-isogeny Selmer groups for Mordell elliptic curves, which in turn relies on the Davenport-Heilbronn theorem on 3-torsion of class groups of quadratic fields. See [2] for the exact computation of the average size of the 3-isogeny Selmer groups of Mordell elliptic curves.…”

Section: Introductionmentioning

confidence: 99%

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Elliptic Curves with Long Arithmetic Progressions Have Large Rank

Garcia-Fritz

Pasten

2020

International Mathematics Research Notices

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For any family of elliptic curves over the rational numbers with fixed j-invariant, we prove that the existence of a long sequence of rational points whose x-coordinates form a non-trivial arithmetic progression implies that the Mordell-Weil rank is large, and similarly for y-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond's quantitative extension of results of Faltings.

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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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The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

Cited by 20 publications

References 36 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

Elliptic Curves with Long Arithmetic Progressions Have Large Rank

Arithmetic statistics of Prym surfaces

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