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Published October 2020 | public
Journal Article

Quantitative non-vanishing of Dirichlet L-values modulo p


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,λ_χ) 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_ℓ-extension of Q for an odd prime ℓ≠p with the corresponding critical L-value twists being p-indivisible.

Additional Information

© 2020 Springer Verlag. Received 27 January 2020; Revised 18 May 2020; Published 09 July 2020. 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. Communicated by Kannan Soundararajan.

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August 19, 2023
October 20, 2023