Twisted adjoint L-values, dihedral congruence primes and the Bloch–Kato conjecture

Abstract

We show that a dihedral congruence prime for a normalised Hecke eigenform f in \(S_k(\Gamma _0(D),\chi _D)\), where \(\chi _D\) is a real quadratic character, appears in the denominator of the Bloch–Kato conjectural formula for the value at 1 of the twisted adjoint L-function of f. We then use a formula of Zagier to prove that it appears in the denominator of a suitably normalised \(L(1,{\mathrm {ad}}^0(g)\otimes \chi _D)\) for some \(g\in S_k(\Gamma _0(D),\chi _D)\).