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The exceptional zero conjecture for Hilbert modular forms

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 January 2009

Chung Pang Mok
Chung Pang Mok*
Affiliation:
970 Evans, University of California, Berkeley, CA 94720-3840, USA (email: [email protected])
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Abstract

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Using a p-adic analogue of the convolution method of Rankin–Selberg and Shimura, we construct the two-variable p-adic L-function of a Hida family of Hilbert modular eigenforms of parallel weight. It is shown that the conditions of Greenberg–Stevens [R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 407–447] are satisfied, from which we deduce special cases of the Mazur–Tate–Teitelbaum conjecture in the Hilbert modular setting.

Keywords

Hilbert modular formsp-adic L-functionsexceptional zero conjecture

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 1 , January 2009 , pp. 1 - 55
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Blasius, D., Hilbert modular forms and the Ramanujan conjecture, Noncommutative geometry and number theory, Aspects of Mathematics, vol. E37 (Vieweg, Wiesbaden, 2006), 35–56.CrossRefGoogle Scholar
[2]Blasius, D. and Rogawski, J., Motives for Hilbert modular forms, Invent. Math. 114 (1993), 55–87.CrossRefGoogle Scholar
[3]Carayol, H., Sur la mauvaise réduction des courbes de Shimura, Compositio Math. (1986a), 151–230.Google Scholar
[4]Carayol, H., Sur les représentations P-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Ec. Norm. Super., IV. Ser. 19 (1986b), 409–468.CrossRefGoogle Scholar
[5]Dabrowski, A., p-adic L-functions of Hilbert modular forms, Ann. Inst. Fourier (Grenoble) 44 (1994), 1025–1041.CrossRefGoogle Scholar
[6]Dabrowski, A. and Delbourgo, D., S-adic L-functions attached to the symmetric square of a newform, Proc. London Math. Soc. (3) 74 (1997), 559–611.CrossRefGoogle Scholar
[7]Deligne, P. and Ribet, K., Values of abelian L functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227–286.CrossRefGoogle Scholar
[8]Greenberg, R., Trivial zeros of p-adic L-functions, p-adic Monodromy and the Birch and Swinnerton-Dyer conjecture, Contemporary Mathematics, vol. 165 (American Mathematical Society, Providence, RI, 1994), 183–211.CrossRefGoogle Scholar
[9]Greenberg, R. and Stevens, G., p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 407–447.CrossRefGoogle Scholar
[10]Greenberg, R. and Stevens, G., On the conjecture of Mazur, Tate, and Teitelbaum, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, Contemporary Mathematics, vol. 165 (American Mathematical Society, Providence, RI, 1994), 183–211.CrossRefGoogle Scholar
[11]Gross, B., On the factorization of p-adic L-series, Invent. Math. 57 (1980), 83–95.CrossRefGoogle Scholar
[12]Hida, H., On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math. (2) 128 (1988), 295–384.CrossRefGoogle Scholar
[13]Hida, H., On nearly ordinary Hecke algebras for GL(2) over totally real fields, in Algebraic Number Theory, Advanced Studies in Pure Mathematics, vol. 17 (Academic Press, Boston, MA, 1989), 139–169.Google Scholar
[14]Hida, H., On p-adic L-functions of GL(2) times GL(2) over totally real fields, Ann. Inst. Fourier (Grenoble) 41 (1991), 311–391.CrossRefGoogle Scholar
[15]Hida, H., ℒ-invariants of Tate curves, Tate anniversary volume from Pure and Applied Math Quarterly, 5 (2009), to appear.CrossRefGoogle Scholar
[16]Hida, H., Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
[17]Hida, H., Hilbert modular forms and Iwasawa theory (The Clarendon Press/Oxford University Press, Oxford, 2006).CrossRefGoogle Scholar
[18]Mazur, B., Tate, J. and Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1–48.CrossRefGoogle Scholar
[19]Mok, C. P., The exceptional zero conjecture for Hilbert modular forms, Harvard University Thesis (2007) (available at http://math.berkeley.edu/∼mok).Google Scholar
[20]Nagata, M., Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13 (John Wiley & Sons, New York, 1962).Google Scholar
[21]Panchishkin, A., Convolutions of Hilbert modular forms and their non-Archimedean analogues, Math. USSR-Sb. 64 (1989), 571–584.CrossRefGoogle Scholar
[22]Panchishkin, A., Non-Archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Mathematics, vol. 1471 (Springer, Berlin, 1991).CrossRefGoogle Scholar
[23]Panchishkin, A., Two variable p-adic L functions attached to eigenfamilies of positive slope, Invent. Math. 154 (2003), 551–615.CrossRefGoogle Scholar
[24]Rohrlich, D., On L-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 409–423.CrossRefGoogle Scholar
[25]Rohrlich, D., L-functions and division towers, Math. Ann. 281 (1988), 611–632.CrossRefGoogle Scholar
[26]Rohrlich, D., Nonvanishing of L-functions for GL(2), Invent. Math. 97 (1989), 381–403.CrossRefGoogle Scholar
[27]Grothendieck, A, Groupes de monodromie en géométrie algébrique, Séminaire de géométrie algébrique, Lecture Notes in Mathematics, vol. 288 (Springer, Berlin, 1972).Google Scholar
[28]Shimura, G., The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), 637–679.CrossRefGoogle Scholar
[29]Shimura, G., On the Eisenstein series of Hilbert modular groups, Revista Mat. Iberoamer. 1 (1985), 1–42.CrossRefGoogle Scholar
[30]Shimura, G., Arithmeticity in the theory of automorphic forms (American Mathematical Society, Providence, RI, 2000).Google Scholar
[31]Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265–280.CrossRefGoogle Scholar
[32]Washington, L., Introduction to cyclotomic fields (Springer, New York, 1997).CrossRefGoogle Scholar
[33]Wiles, A., On ordinary λ-adic representations associated to modular forms, Invent. Math. 94 (1988), 529–573.CrossRefGoogle Scholar