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arXiv · 1910.12798

Arithmetic conjectures suggested by the statistical behavior of modular symbols

Barry Mazur·Karl Rubin
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Abstract

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. 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 $\mathbb{Q}$. 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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Barry Mazur, Karl Rubin. 2019-10-28. Arithmetic conjectures suggested by the statistical behavior of modular symbols. https://arxiv.org/abs/1910.12798

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