Exceptional zeros and L-invariants of Bianchi modular forms
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European Commisision's projectBSD - Euler systems and the conjectures of Birch and Swinnerton-Dyer, Bloch and Kato (EC-H2020-682152)
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.
CitationBarrera, D., Williams, C. Exceptional zeros and L-invariants of Bianchi modular forms. "Transactions of the American Mathematical Society", 2018, p. 1-36.