Functoriality of automorphic L-invariants and applications

TL;DR

This paper investigates the behavior of automorphic L-invariants under base change and lifts, providing new proofs of key conjectures in the theory of modular elliptic curves using automorphic and arithmetic methods.

Contribution

It establishes the functoriality of automorphic L-invariants under certain lifts and base changes, leading to a new proof of the exceptional zero conjecture for modular elliptic curves.

Findings

Automorphic and arithmetic L-invariants are equal under specified conditions.

New proof of the exceptional zero conjecture for modular elliptic curves.

Functoriality of automorphic L-invariants under base change and lifts.

Abstract

We study the behaviour of automorphic L-Invariants associated to cuspidal representations of GL(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic L-functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic L-invariants. This together with Spiess' results on p-adic L-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves - at least, up to sign.