Exact double averages of twisted L-values

TL;DR

This paper derives exact formulas for double averages of twisted central L-values of modular forms and ideal class characters, extending stability results and nonvanishing criteria to more general levels and ramification scenarios.

Contribution

It introduces new exact formulas for double averages of twisted L-values, including non-squarefree levels and ramification, generalizing previous stable average results.

Findings

Exact formulas for double averages of L-values

Stability of averages across all ranges under certain conditions

Effective nonvanishing results for central L-values

Abstract

Consider central $L$-values of even weight elliptic or Hilbert modular forms $f$ twisted by ideal class characters $\chi$ of an imaginary quadratic extension $K$. Fixing $\chi$, and assuming $K$ is inert at each prime dividing the level, one knows simple exact formulas for averages over newforms $f$ of squarefree levels satisfying a parity condition on the number of prime factors. These averages stabilize when the level is large with respect to $K$ (the "stable range"). In weight 2, we obtain exact formulas for a simultaneous average over both $f$ and $\chi$. We allow for non-squarefree levels with any number of prime factors, and ramification or splitting of $K$ above the level. Under elementary conditions on the level, these double averages are "stable" in all ranges. Two consequences are generalizations of the aforementioned stable (single) averages and effective results on nonvanishing of central $L$-values.