Hilbert modular forms and the Gross-Stark conjecture. (English) Zbl 1250.11099

This paper concerns B. H. Gross’s \(p\)-adic analogue of the Stark conjecture, introduced in [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 979–994 (1981; Zbl 0507.12010)]. This conjecture describes the derivatives at \(s = 0\) of \(p\)-adic \(L\)-functions attached to totally odd ideal class characters \(\chi\) of totally real number fields \(F\), in cases where \(\chi(\mathfrak{p}) = 1\) for some prime \(\mathfrak{p} \mid p\) (in which case the value of the \(p\)-adic \(L\)-function at \(s = 0\) is forced to vanish, because of the form of the interpolating factor at \(p\)). Gross’s conjecture predicts that the derivative of the \(p\)-adic \(L\)-function is equal to the value of the complex \(L\)-function multiplied by an “\(\mathcal{L}\)-invariant” defined in terms of a certain \(p\)-unit of the splitting field of \(\chi\). It was shown in the original paper of Gross that the conjecture holds when \(F = \mathbb{Q}\).
This paper proves this conjecture for a wide class of \(F\) and \(\chi\). More specifically, the conjecture is proved under the assumption that (i) Leopoldt’s conjecture holds for \(F\), and either (ii) there are at least two primes of \(F\) above \(p\) or (ii’) a certain condition relating the \(L\)-invariants of \(\chi\) and \(\chi^{-1}\) holds.
For a sketch of the argument, the reader is referred to the paper’s very clear and informative introduction.


11R42 Zeta functions and \(L\)-functions of number fields
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F85 \(p\)-adic theory, local fields


Zbl 0507.12010
Full Text: DOI


[1] J. Coates and S. Lichtenbaum, ”On \(l\)-adic zeta functions,” Ann. of Math., vol. 98, pp. 498-550, 1973. · Zbl 0279.12005 · doi:10.2307/1970916
[2] P. Colmez, ”Résidu en \(s=1\) des fonctions zêta \(p\)-adiques,” Invent. Math., vol. 91, iss. 2, pp. 371-389, 1988. · Zbl 0651.12010 · doi:10.1007/BF01389373
[3] P. Colmez, ”Fonctions zêta \(p\)-adiques en \(s=0\),” J. Reine Angew. Math., vol. 467, pp. 89-107, 1995. · Zbl 0864.11062 · doi:10.1515/crll.1995.467.89
[4] P. Deligne and K. A. Ribet, ”Values of abelian \(L\)-functions at negative integers over totally real fields,” Invent. Math., vol. 59, iss. 3, pp. 227-286, 1980. · Zbl 0434.12009 · doi:10.1007/BF01453237
[5] B. Ferrero and R. Greenberg, ”On the behavior of \(p\)-adic \(L\)-functions at \(s=0\),” Invent. Math., vol. 50, iss. 1, pp. 91-102, 1978/79. · Zbl 0441.12003 · doi:10.1007/BF01406470
[6] R. Greenberg, ”On \(p\)-adic Artin \(L\)-functions,” Nagoya Math. J., vol. 89, pp. 77-87, 1983. · Zbl 0513.12012
[7] R. Greenberg, ”Trivial zeros of \(p\)-adic \(L\)-functions,” in \(p\)-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, Providence, RI, 1994, pp. 149-174. · Zbl 0838.11070
[8] R. Greenberg, ”Introduction to Iwasawa theory for elliptic curves,” in Arithmetic Algebraic Geometry, Providence, RI, 2001, pp. 407-464. · Zbl 1002.11048
[9] B. H. Gross, ”\(p\)-adic \(L\)-series at \(s=0\),” J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 28, iss. 3, pp. 979-994 (1982), 1981. · Zbl 0507.12010
[10] N. M. Katz, ”\(p\)-adic \(L\)-functions for CM fields,” Invent. Math., vol. 49, iss. 3, pp. 199-297, 1978. · Zbl 0417.12003 · doi:10.1007/BF01390187
[11] T. Miyake, Modular Forms, New York: Springer-Verlag, 1989. · Zbl 0701.11014
[12] K. A. Ribet, ”A modular construction of unramified \(p\)-extensions of \(\mathbfQ(\mu_p)\),” Invent. Math., vol. 34, iss. 3, pp. 151-162, 1976. · Zbl 0338.12003 · doi:10.1007/BF01403065
[13] G. Shimura, ”The special values of the zeta functions associated with Hilbert modular forms,” Duke Math. J., vol. 45, iss. 3, pp. 637-679, 1978. · Zbl 0394.10015 · doi:10.1215/S0012-7094-78-04529-5
[14] C. L. Siegel, ”Über die Fourierschen Koeffizienten von Modulformen,” Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, vol. 1970, pp. 15-56, 1970. · Zbl 0225.10031
[15] J. Tate, Les Conjectures de Stark sur les Fonctions \(L\) d’Artin en \(s=0\), Lecture notes edited by Dominique Bernardi and Norbert Schappacher, Boston, MA: Birkhäuser Boston Inc., 1984, vol. 47. · Zbl 0545.12009
[16] A. Wiles, ”On \(p\)-adic representations for totally real fields,” Ann. of Math., vol. 123, iss. 3, pp. 407-456, 1986. · Zbl 0613.12013 · doi:10.2307/1971332
[17] A. Wiles, ”On ordinary \(\lambda\)-adic representations associated to modular forms,” Invent. Math., vol. 94, iss. 3, pp. 529-573, 1988. · Zbl 0664.10013 · doi:10.1007/BF01394275
[18] A. Wiles, ”The Iwasawa conjecture for totally real fields,” Ann. of Math., vol. 131, iss. 3, pp. 493-540, 1990. · Zbl 0719.11071 · doi:10.2307/1971468
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