Average size of Selmer group in large q limit

2021/01/31 by Park, Sun Woo, Wang, Niudun
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2102.00549

Abstract

In this paper, we prove a function field-analogue of Poonen-Rains heuristics on the average size of p-Selmer group. Let E be an elliptic curve defined over ℤ[t]. Then E is also defined over \mathbbF_q for any q of prime power. We show that for large enough q, the average size of the p-Selmer groups over the family of quadratic twists of E over \mathbbF_q[t] is equal to p+1 for all but finitely many primes p. Namely, if we twist the curve in \mathbbF_q[t] by polynomials of fixed degree n and let both n and q approach to infinity, then the average rank of p-Selmer group converges to p+1.

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