Average size of 2-Selmer groups of elliptic curves over function fields

Ho, Quoc P.; Hùng, V.B. Lê; Ngô, Bảo Châu

doi:10.4310/mrl.2014.v21.n6.a6

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Cited by 4 publications

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The geometric distribution of Selmer groups of elliptic curves over function fields

Feng,

Landesman,

Rains

2022

Math. Ann.

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Fix a positive integer n and a finite field $${\mathbb {F}}_q$$ F q . We study the joint distribution of the rank $${{\,\mathrm{rk}\,}}(E)$$ rk ( E ) , the n-Selmer group $$\text {Sel}_n(E)$$ Sel n ( E ) , and the n-torsion in the Tate–Shafarevich group "Equation missing" as E varies over elliptic curves of fixed height $$d \ge 2$$ d ≥ 2 over $${\mathbb {F}}_q(t)$$ F q ( t ) . We compute this joint distribution in the large q limit. We also show that the “large q, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.

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The geometric distribution of Selmer groups of elliptic curves over function fields

Feng,

Landesman,

Rains

2022

Math. Ann.

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2-Selmer groups of even hyperelliptic curves over function fields

Dao

2023

Trans. Amer. Math. Soc.

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In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg’s representation of the group G = PSO ( 2 n + 2 ) G=\text {PSO}(2n+2) and a Hitchin fibration. Consistent with the result over Q \mathbb {Q} of Arul Shankar and Xiaoheng Wang [Compos. Math. 154 (2018), pp. 188–222], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperelliptic curves, we obtain precisely average number 6.

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Average size of 2-Selmer groups of Jacobians of odd hyperelliptic curves over function fields

Dao

2022

Pacific J. Math.

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Average size of 2-Selmer groups of elliptic curves over function fields

Abstract: Employing a geometric setting inspired by the proof of the Fundamental Lemma, we study some counting problems related to the average size of 2-Selmer groups and hence obtain an estimate for it.

Cited by 4 publications

References 9 publications

The geometric distribution of Selmer groups of elliptic curves over function fields

The geometric distribution of Selmer groups of elliptic curves over function fields

2-Selmer groups of even hyperelliptic curves over function fields

Average size of 2-Selmer groups of Jacobians of odd hyperelliptic curves over function fields

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