Quantitative non-vanishing of Dirichlet L -values modulo p
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Mathematische Annalen
Quantitative non-vanishing of Dirichlet L-values modulo p Ashay Burungale1
· Hae-Sang Sun2
Received: 27 January 2020 / Revised: 18 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let p be an odd prime and k a non-negative integer. Let N be a positive integer such that p N and λ a Dirichlet character modulo N . We obtain quantitative lower bounds for the number of Dirichlet character χ modulo F with the critical Dirichlet L-value L(−k, λχ ) being p-indivisible. Here F → ∞ with (N , F) = 1 and p Fφ(F). We explore the indivisibility via an algebraic and a homological approach. The latter leads to a bound of the form F 1/2 . The p-indivisibility yields results on the distribution of the associated p-Selmer ranks. We also consider an Iwasawa variant. It leads to an explicit upper bound on the lowest conductor of the characters factoring through the Iwasawa Z -extension of Q for an odd prime = p with the corresponding critical L-value twists being p-indivisible. Mathematics Subject Classification Primary 11R18 · 11R29 · 11R42; Secondary 11R23
Communicated by Kannan Soundararajan. A. Burungale and H.-S. Sun are grateful to Haruzo Hida, Barry Mazur and Ye Tian for helpful suggestions. They are also grateful to Philippe Michel and Peter Sarnak for insightful conversations and encouragement. Finally, we are indebted to the referee. The current form of the article owes much to the thorough comments and suggestions of the referee. The research was conceived when the authors were visiting Korea Institute for Advanced Study. They thank the institute for the support and hospitality. H.-S. Sun is supported by the Research Fund (1.150067.01) of UNIST.
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Ashay Burungale [email protected] Hae-Sang Sun [email protected]
1
California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA
2
Ulsan National Institute of Science and Technology, Ulsan, Korea
123
A. Burungale, H.-S. Sun
Contents 1 Introduction . . . . . . . . . . . . . . . . 2 Algebraic expression for Dirichlet L-values 3 Non-vanishing mod p : algebraic setting . 4 Integral expression of Dirichlet L-values . 5 Non-vanishing mod p: homological setting 6 Cyclotomic twists . . . . . . . . . . . . . 7 Numerical examples for r, s, and s . . . . References . . . . . . . . . . . . . . . . . . .
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1 Introduction For a motive M, a critical L-value is a fundamental arithmetic invariant. The vanishing or non-vanishing of the L-value is conjecturally closely related to
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