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arXiv · 2102.03087

Analytic ranks of automorphic L-functions and Landau-Siegel zeros

Hung M. Bui·Kyle Pratt·Alexandru Zaharescu
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Abstract

We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level $q$ have analytic rank $\leq 1$. We show that either Landau-Siegel zeros do not exist, or that almost all such newforms have analytic rank $\leq 2$. In particular, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes $q$ we show the rank of $J_0(q)$ is asymptotically equal to the rank predicted by the Brumer-Murty conjecture.

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Hung M. Bui, Kyle Pratt, Alexandru Zaharescu. 2021-02-05. Analytic ranks of automorphic L-functions and Landau-Siegel zeros. https://arxiv.org/abs/2102.03087

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