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

Hung Bui, Kyle Pratt, Alexandru Zaharescu

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Abstract

We relate the study of Landau-Siegel zeros to the ranks of Jacobians J0(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 ≤ 1. We show that either Landau-Siegel zeros do not exist, or that, for wide ranges of q, almost all such newforms have analytic rank ≤ 2. In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes q in a wide range we show the rank of J0(q) is asymptotically equal to the rank predicted by the Brumer-Murty conjecture.
Original languageEnglish
Article numbere12834
JournalJournal of the London Mathematical Society
Volume109
Issue number1
Early online date21 Dec 2023
DOIs
Publication statusPublished - 1 Jan 2024

Keywords

  • Landau-Siegel zeros
  • exceptional characters
  • automorphic L-functions
  • analytic ranks
  • ranks of Jacobians
  • Birch and Swinnerton-Dyer conjecture
  • nonvanishing
  • mollfier

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