Exceptional zeros and L-invariants of Bianchi modular forms

Abstract

Let f be a Bianchi modular form, that is, an automorphic form for GL2 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 p of F above p we prove the existence of an Linvariant Lp, depending only on p and f, such that when the p-adic L-function of f has an exceptional zero at p, its derivative can be related to the classical L-value multiplied by Lp. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL2/Q. When p is not split and f is the base-change of a classical modular form f ˜, we relate Lp to the L-invariant of f ˜, resolving a conjecture of Trifkovi´c in this case.