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Secondary terms in the first moment of $|{\rm Sel}_2(E)|$

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arxiv 2412.00995 v1 pith:QGWBBJTJ submitted 2024-12-01 math.NT

Arul Shankar , Takashi Taniguchi This is my paper · ORCID
classification math.NT
keywords termscountingerrorfirstordersecondarytermbhargava--shankar
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abstract

We prove the existence of secondary terms of order $X^{3/4}$, with power saving error terms, in the counting functions of $|{\rm Sel}_2(E)|$, the 2-Selmer group of E, for elliptic curves E having height bounded by X. This is the first improvement on the error term of $o(X^{5/6})$, proved by Bhargava--Shankar, where the primary term of order $X^{5/6}$ for this counting function was obtained.

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  1. Boundedness of average rank of elliptic curves ordered by the coefficients

    math.NT 2025-06 conditional novelty 7.0 of 10

    Ordered by max(|A|,|B|), elliptic curves have average rank at most 1.5, proved via a new semi-invariant counting argument for binary quartic forms.

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