Central derivatives of L-functions in Hida families

TL;DR

This paper proves that within a Hida family of modular forms, the order of vanishing of L-functions at s=1 is consistent across almost all forms, linking the behavior of special values to p-adic L-functions.

Contribution

It establishes a new uniformity result for the order of vanishing of L-functions in Hida families, extending previous understanding of their behavior.

Findings

Order of vanishing one at s=1 propagates to almost all forms in the family.

Order of vanishing zero case follows from the Mazur-Kitagawa p-adic L-function.

Results connect special L-values with p-adic L-functions in Hida families.

Abstract

We prove a result of the following type: given a Hida family of modular forms, if there exists a weight two form in the family whose L-function vanishes to exact order one at s=1, then all but finitely many weight two forms in the family enjoy this same property. The analogous result for order of vanishing zero is also true, and is an easy consequence of the existence of the Mazur-Kitagawa two-variable p-adic L-function.