Functoriality of Automorphic $\mathrm{L}$-Invariants and Applications

Abstract

We study the behaviour of automorphic $\mathrm{L}$-invariants associated to cuspidal representations of $\mathrm{GL}(2)$ 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 $\mathrm{L}$-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 $\mathrm{L}$-invariants. This together with Spieß' results on $p$-adic $\mathrm{L}$-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves -- at least, up to sign.