# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.00619