The second moment of the size of the $2$-Selmer group of elliptic curves

Arul Shankar; Ashvin Swaminathan; Manjul Bhargava

The second moment of the size of the $2$-Selmer group of elliptic curves

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arxiv 2110.09063 v1 pith:N6YVHBEJ submitted 2021-10-18 math.NT

Manjul Bhargava , Arul Shankar , Ashvin Swaminathan This is my paper

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keywords curvesellipticgroupmomentsecondselmersizeconfirms

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abstract

In this paper, we prove that when elliptic curves over $\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains.

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Tamagawa ratios and unbounded Selmer moments
math.NT 2026-06 unverdicted novelty 6.0 of 10

Framework gives conjectural characterization of geometric families of abelian varieties with unbounded average l-Selmer sizes, proven correct when l-torsion module is constant across the family.

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