On the critical values of certain Dirichlet series and the periods of automorphic forms. (English) Zbl 0656.10018

Arithmeticity questions involving special (critical) values of zeta functions (Dirichlet series) associated with automorphic forms of the Hilbert modular type (or like Eisenstein series) or concerning periods of automorphic forms (and, in particular, of abelian integrals on arithmetic curves) have, recently for several years, been deeply investigated in a series of papers by the author. The present important paper is yet another milestone in the same direction, offering not only generalizations of some of his earlier arithmeticity theorems (now, without parity conditions or other exceptions and covering the ‘perpetually present’ forms of half-integral weight, too) but also conjectures expected to have far-reaching consequences and open up ‘new territory’.
Reviewer: S.Raghavan


11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F27 Theta series; Weil representation; theta correspondences
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F33 Congruences for modular and \(p\)-adic modular forms
Full Text: DOI EuDML


[1] [B] Bluher, A.: Thesis, Princeton University, 1988
[2] [H] Hida, H.: On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves. Am. J. Math.103, 727-776 (1981) · Zbl 0477.14024
[3] [I] Im, J.: Thesis, Princeton University, 1988
[4] [S] Shimizu, H.: On zeta functions of quarternion algebras. Ann. Math.81, 166-193 (1965) · Zbl 0201.37903
[5] [59] Shimura, G.: Sur les intégrales attachées aux formes automorphes. J. Math. Soc. Japan11, 291-311 (1959) · Zbl 0090.05503
[6] [62] Shimura, G.: On Dirichlet series and abelian varieties attached to automorphic forms. Ann. Math.76, 237-294 (1962) · Zbl 0142.05501
[7] [67] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58-159 (1967) · Zbl 0204.07201
[8] [71] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. (Publ. Math. Soc. of Japan, Vol. 11). Iwanami Shoten and Princeton Univ. Press, 1971 · Zbl 0221.10029
[9] [73] Shimura, G.: On the factors of the jacobian variety of a modular function field. J. Math. Soc. Japan25, 523-544 (1973) · Zbl 0266.14017
[10] [75] Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. London Math. Soc. (Ser. 3)31, 79-98 (1975) · Zbl 0311.10029
[11] [77] Shimura, G.: On the periods of modular forms. Math. Ann.229, 211-221 (1977) · Zbl 0363.10019
[12] [78] Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J.45, 637-679 (1978) · Zbl 0394.10015
[13] [80] Shimura, G.: The arithmetic of certain zeta functions and automorphic forms on orthogonal groups. Ann. Math.111, 313-375 (1980) · Zbl 0438.12003
[14] [81] Shimura, G.: On certain zeta functions attached to two Hilbert modular forms I, II. Ann. Math.114, 127-164, 569-607 (1981) · Zbl 0468.10016
[15] [81z] Shimura, G.: The critical values of certain zeta functions associated with modular forms of halfintegral weight. J. Math. Soc. Japan33, 649-672 (1981) · Zbl 0494.10018
[16] [82] Shimura, G.: The periods of certain automorphic forms of arithmetic type. J. Fac. Sci. Univ. Tokyo (Sec. IA)28, 605-632 (1982) · Zbl 0499.10027
[17] [83a] Shimura, G.: Algebraic relations between critical values of zeta functions and inner products. Am. J. Math.104, 253-285 (1983) · Zbl 0518.10032
[18] [83b] Shimura, G.: On Eisenstein series. Duke Math. J.50, 417-476 (1983) · Zbl 0519.10019
[19] [84] Shimura, G.: Differential operators and the singular values of Eisenstein series. Duke Math. J.51, 261-329 (1984) · Zbl 0546.10025
[20] [85a] Shimura, G.: On Eisenstein series of half-integral weight. Duke Math. J.52, 281-324 (1985) · Zbl 0577.10025
[21] [85b] Shimura, G.: On the Eisenstein series of Hilbert modular groups. Rev. Mat. Ib.1 (No. 3) 1-42 (1985) · Zbl 0608.10028
[22] [86] Shimura, G.: On a class of nearly holomorphic automorphic forms. Ann. Math.123, 347-406 (1986) · Zbl 0593.10022
[23] [87a] Shimura, G.: Nearly holomorphic functions on hermitian symmetric spaces. Math. Ann.278, 1-28 (1987) · Zbl 0636.10023
[24] [87b] Shimura, G.: On Hilbert modular forms of half-integral weight. Duke Math. J.55, 765-838 (1987) · Zbl 0636.10024
[25] [St] Sturm, J.: Special values of zeta functions and Eisenstein series of half integral weight. Am. J. Math.102, 219-240 (1980); Addendum, ibid. pp. 781-783 · Zbl 0433.10015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.