Arithmetic conjectures suggested by the statistical behavior of modular symbols

2019/10/28 by Barry Mazur, Karl Rubin, Mazur, Barry +1 · 1 citation
Mathematics · #11F67 #11G05 #11G40 #14G05 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Benford’s Law and Fraud Detection #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1910.12798

openalex publication_date 2019/10/28 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

Suppose E is an elliptic curve over ℚ and χ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1) = 0. Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell-Weil rank grows in abelian extensions of ℚ. Using this heuristic we find a large class of infinite abelian extensions F where we expect E(F) to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, and Kisilevsky. Where our predictions and theirs overlap, the predictions are consistent.

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