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

Abstract:

Let $f$ be a Bianchi modular form, that is, an automorphic form for $\mathrm {GL}_2$ 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 $\mathcal {L}$-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 $\mathrm {GL}_2/\mathbb {Q}$. When $p$ is not split and $f$ is the base-change of a classical modular form $\tilde {f}$, we relate $\mathcal {L}_{\mathfrak {p}}$ to the $\mathcal {L}$-invariant of $\tilde {f}$, resolving a conjecture of Trifković in this case.