$\mathcal{L}$-invariants and exceptional zeros of Bianchi modular forms

Abstract

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime $\mathfrak{p}$ of F above p we prove the existence of an L-invariant ${\mathcal{L}}_{\mathfrak{p}}$, depending only on $\mathfrak{p}$ and f, such that when the p-adic L-function of f has an exceptional zero at $\mathfrak{p}$, its derivative can be related to the classical L-value multiplied by ${\mathcal{L}}_{\mathfrak{p}}$. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL(2) over the rationals. When p is not split and f is the base-change of a classical modular form F, we relate ${\mathcal{L}}_{\mathfrak{p}}$ to the L-invariant of F, resolving a conjecture of Trifković in this case.

36 Pages; minor updates and corrections. To appear in Transactions of the AMS

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