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# 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

#### 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