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The average number of elements in the 4-Selmer groups of elliptic curves is 7
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The average number of elements in the 4-Selmer groups of elliptic curves is 7
abstract
We prove that when all elliptic curves over $\mathbb{Q}$ are ordered by height, the average size of their 4-Selmer groups is equal to 7. As a consequence, we show that a positive proportion (in fact, at least one fifth) of all 2-Selmer elements of elliptic curves, when ordered by height, do not lift to 4-Selmer elements, and thus correspond to nontrivial 2-torsion elements in the associated Tate--Shafarevich groups.
Forward citations
Cited by 4 Pith papers
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Geometry-of-numbers methods over global fields II: Coregular representations
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Exact classification of elliptic curves $y^{2}=x^{3}-pqx$ with rank $0$ and trivial $\Sha[2]$
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Tamagawa ratios and unbounded Selmer moments
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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Selmer groups of families of elliptic curves with an $\ell$-isogeny
In many families of elliptic curves with a rational prime-degree isogeny, the logarithmic Tamagawa ratio satisfies a central limit theorem, yielding curves with arbitrarily large ℓ-Selmer groups for ℓ = 2, 3, 5, 7, 13.
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