Affiliation:
1. Department of Mathematics University of Manchester Manchester UK
2. All Souls College Oxford UK
3. Department of Mathematics University of Illinois at Urbana‐Champaign Urbana, Illinois USA
4. Simion Stoilow Institute of Mathematics of the Romanian Academy Bucharest Romania
Abstract
AbstractWe relate the study of Landau–Siegel zeros to the ranks of Jacobians of modular curves for large primes . 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 have analytic rank . We show that either Landau–Siegel zeros do not exist, or that, for wide ranges of , almost all such newforms have analytic rank . In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes in a wide range, we show that the rank of is asymptotically equal to the rank predicted by the Brumer–Murty conjecture.