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 \(g\in S_k(\Gamma _0(D),\chi _D)\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bloch, S., Kato, K.: \(L\)-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift Volume I, pp. 333–400, Progress in Mathematics, 86, Birkhäuser, Boston, (1990)
Brown, A.F., Ghate, E.P.: Dihedral congruence primes and class fields of real quadratic fields. J. Number Theory 95, 14–37 (2002)
Deligne, P.: Valeurs de Fonctions \(L\) et Périodes d’Intégrales. In: AMS Proc. Symp. Pure Math., vol. 33, part 2, 313–346 (1979)
Deligne, P.: Formes modulaires et représentations \(l\)-adiques. Sém. Bourbaki, éxp. 355. In: Lecture Notes in Mathematics, vol. 179, pp. 139–172. Springer, Berlin (1969)
Diamond, F., Flach, M., Guo, L.: The Tamagawa number conjecture of adjoint motives of modular forms. Ann. Sci. École Norm. Sup. 37(4), 663–727 (2004)
Diamond, F., Flach, M., Guo, L.: The Bloch–Kato conjecture for adjoint motives of modular forms. Math. Res. Lett. 8, 437–442 (2001)
Doi, K., Hida, H., Ishii, H.: Discriminant of Hecke fields and twisted adjoint \(L\)-values for GL(2). Invent. math 134, 547–577 (1998)
Dummigan, N.: Symmetric square \(L\)-functions and Shafarevich–Tate groups, II. Int. J. Number Theory 5, 1321–1345 (2009)
Dummigan, N., Heim, B.: Symmetric square \(L\)-values and dihedral congruences for cusp forms. J. Number Theory 130, 2078–2091 (2010)
Faltings, G., Jordan, B.: Crystalline cohomology and \(GL(2,\mathbb{Q})\). Israel J. Math. 90, 1–66 (1995)
Ghate, E. P.: Congruences between base-change and non-base-change Hilbert modular forms. In: Cohomology of arithmetic groups, L-functions and automorphic forms (Mumbai, 1998/1999), pp. 35–62, Tata Inst. Fund. Res. Stud. Math., 15, Tata Inst. Fund. Res., Bombay (2001)
Hida, H.: Global quadratic units and Hecke algebras. Doc. Math. 3, 273–284 (1998)
Hida, H.: Geometric Modular Forms and Elliptic Curves. World Scientific Publishing Co. Inc., River Edge, NJ (2000)
Hida, H.: Modular forms and Galois cohomology. Cambridge University Press, Cambridge (2000)
Koike, M.: Congruences between cusp forms and linear representations of the Galois group. Nagoya Math. J. 64, 63–85 (1976)
Ohta, M.: The representation of Galois group attached to certain finite group schemes, and its application to Shimura’s theory, pp. 149–156 in Algebraic Number Theory, Proc. Int. Symp. Kyoto, : Japan Soc, p. 1977. Prom. Sci, Tokyo (1976)
Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton Univ Press, Princeton (1971)
Tilouine, J., Urban, E.: Integral period relations and congruences, preprint 2018. http://www.math.columbia.edu/~urban/EURP.html
Zagier, D.: Modular forms whose coefficients involve zeta-functions of quadratic fields, Modular functions of one variable VI, Lecture Notes in Mathematics 627, pp. 105–169, Springer (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jens Funke.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-020-00224-w