Vanishing of special values and central derivatives in Hida families

TL;DR

This paper investigates the relationship between the order of vanishing of twisted L-functions at the central point and the rank of associated Selmer groups within Hida families, extending previous results to more general settings.

Contribution

It establishes a link between the order of vanishing of twisted L-functions and the rank of Nekovář-Selmer groups in Hida families over quadratic fields, generalizing prior work.

Findings

If a twisted L-function has order of vanishing r ≤ 1, then the Selmer group's rank equals r.

Infinitely many twisted L-functions in the Hida family share the same order of vanishing.

Results extend Howard's work to broader arithmetic contexts involving quadratic fields.

Abstract

The theme of this work is the study of the Nekov\'a\v{r}-Selmer group H^1_f(K,T) attached to a twisted Hida family T of Galois representations and a quadratic number field K. The results that we obtain have the following shape: if a twisted L-function of a suitable modular form in the Hida family has order of vanishing r at most 1 at the central critical point then the rank of H^1_f(K,T) as a module over a certain local Hida-Hecke algebra is equal to r. Under the above assumption, we also show that infinitely many twisted L-functions of modular forms in the Hida family have the same order of vanishing at the central critical point. Our theorems extend to more general arithmetic situations results obtained by Howard when K is an imaginary quadratic field and all the primes dividing the tame level of the Hida family split in K.