Mathematical Research Letters

Volume 26 (2019)

Number 4

Non-vanishing of Hecke $L$-functions and Bloch–Kato $p$-Selmer groups

Pages: 1145 – 1177

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n4.a8

Authors

Arianna Iannuzzi (Department of Mathematics and Computer Science, Saint Mary’s College, Notre Dame, Indiana, U.S.A.)

Byoung Du Kim (School of Mathematics and Statistics, Victoria University of Wellington, New Zealand)

Riad Masri (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Alexander Mathers (Department of Mathematics, University of California at San Diego)

Maria Ross (Department of Mathematics and Computer Science, University of Puget Sound, Tacoma, Washington, U.S.A.)

Wei-Lun Tsai (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Abstract

The canonical Hecke characters in the sense of Rohrlich form a set of algebraic Hecke characters with important arithmetic properties. In this paper, we prove that for an asymptotic density of 100% of imaginary quadratic fields $K$ within certain general families, the number of canonical Hecke characters of $K$ whose $L$-function has a nonvanishing central value is $\gg {\lvert \mathrm{disc} (K) \rvert}^{\delta}$ for some absolute constant $\delta \gt 0$. We then prove an analogous density result for the number of canonical Hecke characters of $K$ whose associated Bloch–Kato $p$-Selmer group is finite. Among other things, our proofs rely on recent work of Ellenberg, Pierce, and Wood on bounds for torsion in class groups, and a careful study of the main conjecture of Iwasawa theory for imaginary quadratic fields.

The authors were partially supported by the NSF grants DMS-1162535 and DMS-1460766 and the Simons Foundation grant #421991 while completing this work.

Received 15 December 2017

Accepted 31 July 2018

Published 25 October 2019