Non-vanishing of critical $L$-values in Hida families
Abstract
We study the vanishing of $L(f,\chi,j)$ as $f$ runs through all classical forms in a $p$-adic Hida family (including forms with arbitrarily high nebentype at $p$), $\chi$ runs through all characters of $p$-power conductor, and $j$ is a critical value. We show that if infinitely many of these $L$-values vanish (apart from the ones forced to vanish by the sign of their functional equation) then this infinitude of vanishing must be exceptionally regular, so regular in fact that one can typically rule out this possibility in any given example. Indeed, we systematically verified that such regular vanishing does not occur in multiple Hida families twisted by a wide range of quadratic characters by computing the corresponding two-variable $p$-adic $L$-functions via overconvergent modular symbols.
Fields of Study & Subjects
advanced mathematical theories
Advanced Algebra and Geometry
Algebraic Geometry and Number Theory
Publication details
- Journal / source
- Mathematical Research Letters
- Publisher
- International Press of Boston
- Volume
- 33
- Issue
- 1
- Pages
- 57–81
- Publication date
- 1 January 2026
- Language
- EN