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 \(g\in S_k(\Gamma _0(D),\chi _D)\).
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Communicated by Jens Funke.
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Dummigan, N. Twisted adjoint L-values, dihedral congruence primes and the Bloch–Kato conjecture. Abh. Math. Semin. Univ. Hambg. 90, 215–227 (2020). https://doi.org/10.1007/s12188-020-00224-w
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DOI: https://doi.org/10.1007/s12188-020-00224-w