On the fundamental periods of Hilbert modular forms

Dou, Ze-Li

doi:10.1090/S0002-9947-1994-1264144-9

On the fundamental periods of Hilbert modular forms
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Trans. Amer. Math. Soc. 346 (1994), 147-158 Request permission

Abstract:

The main purpose of this paper is to establish the existence of fundamental periods of primitive cusp forms of Hilbert modular type of several variables, as well as the relationship between those fundamental periods and the special values of the associated $L$-functions. These results, together with some recent results of Shimura, give us the means of translating with ease results concerning periods of automorphic forms derived from various points of view. We also verify several conjectures of Shimura on the properties of such fundamental periods.

References

Armand Borel, Stable real cohomology of arithmetic groups. II, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 21–55. MR 642850
Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917

Fundamental periods of certain arithmetic cusp forms

G. Harder, Eisenstein cohomology of arithmetic groups. The case $\textrm {GL}_2$, Invent. Math. 89 (1987), no. 1, 37–118. MR 892187, DOI 10.1007/BF01404673
Haruzo Hida, On the critical values of $L$-functions of $\textrm {GL}(2)$ and $\textrm {GL}(2)\times \textrm {GL}(2)$, Duke Math. J. 74 (1994), no. 2, 431–529. MR 1272981, DOI 10.1215/S0012-7094-94-07417-6
Yozô Matsushima and Goro Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. of Math. (2) 78 (1963), 417–449. MR 155340, DOI 10.2307/1970534
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
Goro Shimura, On the critical values of certain Dirichlet series and the periods of automorphic forms, Invent. Math. 94 (1988), no. 2, 245–305. MR 958833, DOI 10.1007/BF01394326
Goro Shimura, On the fundamental periods of automorphic forms of arithmetic type, Invent. Math. 102 (1990), no. 2, 399–428. MR 1074480, DOI 10.1007/BF01233433