On special zeros of $p$-adic $L$-functions of Hilbert modular forms

TL;DR

This paper proves a conjecture linking derivatives of $p$-adic $L$-functions of elliptic curves over totally real fields to $p$-adic periods, advancing understanding of special zeros in the context of Hilbert modular forms.

Contribution

It establishes the weak exceptional zero conjecture and derives the strong form under mild conditions, introducing a new local data-based construction of the $p$-adic $L$-function.

Findings

Proved the weak exceptional zero conjecture for elliptic curves over totally real fields.

Derived the strong exceptional zero conjecture from a special case by Mok.

Introduced a novel construction of $p$-adic $L$-functions using local data.

Abstract

Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to the corresponding Hilbert modular form at the places above $p$ where $E$ has split multiplicative reduction. Under some mild restrictions on $p$ and the conductor of $E$ we deduce the exceptional zero conjecture in the strong form (i.e.\ where the automorphic $p$-adic periods are replaced by the $\cL$-invariants of $E$ defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the $p$-adic $L$-function of $E$ in terms of local data.