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

Hung M. Bui, Kyle Pratt, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


We relate the study of Landau–Siegel zeros to the ranks of Jacobians (Formula presented.) of modular curves for large primes (Formula presented.). 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 (Formula presented.) have analytic rank (Formula presented.). We show that either Landau–Siegel zeros do not exist, or that, for wide ranges of (Formula presented.), almost all such newforms have analytic rank (Formula presented.). In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes (Formula presented.) in a wide range, we show that the rank of (Formula presented.) is asymptotically equal to the rank predicted by the Brumer–Murty conjecture.

Original languageEnglish (US)
Article numbere12834
JournalJournal of the London Mathematical Society
Issue number1
StatePublished - Jan 2024

ASJC Scopus subject areas

  • General Mathematics


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