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

TL;DR

This paper explores the connection between Landau-Siegel zeros and the ranks of Jacobians of modular curves, providing evidence that either such zeros do not exist or most newforms have low analytic rank.

Contribution

It establishes a link between Landau-Siegel zeros and the ranks of Jacobians, showing that almost all newforms have analytic rank at most 2, supporting the Brumer-Murty conjecture.

Findings

Almost all odd newforms have analytic rank equal to one.

Either Landau-Siegel zeros do not exist, or most newforms have analytic rank ≤ 2.

For certain primes, the rank of J_0(q) matches the Brumer-Murty conjecture predictions.

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.