Non-vanishing of critical $L$-values in Hida families

TL;DR

This paper investigates the conditions under which critical L-values in Hida families vanish infinitely often, demonstrating that such regular vanishing is highly unlikely and providing computational evidence against it.

Contribution

The paper establishes that infinite regular vanishing of critical L-values in Hida families is extremely rare and verifies this through extensive computational analysis.

Findings

Regular vanishing of L-values is highly unlikely in Hida families.

Computational evidence shows no such regular vanishing occurs in tested cases.

Two-variable p-adic L-functions can be effectively computed using overconvergent modular symbols.

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.