A note on a lemma of Shimura

Author:
Ze-Li Dou

Journal:
Proc. Amer. Math. Soc. **131** (2003), 77-82

MSC (2000):
Primary 11F30, 11F41, 11F55, 11F67

Published electronically:
May 22, 2002

MathSciNet review:
1929025

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The subject of this paper is a quaternion algebra over a quadratic extension of a totally real algebraic number field. We generalize an important technical lemma of Shimura, on which a certain period of automorphic forms depends.

**[D1]**Ze-Li Dou,*On a theta correspondence with respect to a quadratic extension*, J. Number Theory**79**(1999), no. 1, 1–18. MR**1724250**, 10.1006/jnth.1999.2416**[D2]**Ze-Li Dou,*Theta correspondence and Hecke operators relative to a quadratic extension*, Acta Arith.**93**(2000), no. 3, 237–255. MR**1759917****[D3]**Ze-Li Dou,*On the fundamental periods of Hilbert modular forms*, Trans. Amer. Math. Soc.**346**(1994), no. 1, 147–158. MR**1264144**, 10.1090/S0002-9947-1994-1264144-9**[He85]**Dennis A. Hejhal,*A classical approach to a well-known spectral correspondence on quaternion groups*, Number theory (New York, 1983–84) Lecture Notes in Math., vol. 1135, Springer, Berlin, 1985, pp. 127–196. MR**803353**, 10.1007/BFb0074604**[Sh80]**Goro Shimura,*The arithmetic of certain zeta functions and automorphic forms on orthogonal groups*, Ann. of Math. (2)**111**(1980), no. 2, 313–375. MR**569074**, 10.2307/1971202**[Sh81I]**Goro Shimura,*On certain zeta functions attached to two Hilbert modular forms. I. The case of Hecke characters*, Ann. of Math. (2)**114**(1981), no. 1, 127–164. MR**625349**, 10.2307/1971381**[Sh81II]**Goro Shimura,*On certain zeta functions attached to two Hilbert modular forms. II. The case of automorphic forms on a quaternion algebra*, Ann. of Math. (2)**114**(1981), no. 3, 569–607. MR**634429**, 10.2307/1971302**[Sh82]**Goro Shimura,*The periods of certain automorphic forms of arithmetic type*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**28**(1981), no. 3, 605–632 (1982). MR**656039****[Sh88]**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**, 10.1007/BF01394326**[Sh90]**Goro Shimura,*On the fundamental periods of automorphic forms of arithmetic type*, Invent. Math.**102**(1990), no. 2, 399–428. MR**1074480**, 10.1007/BF01233433

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11F30,
11F41,
11F55,
11F67

Retrieve articles in all journals with MSC (2000): 11F30, 11F41, 11F55, 11F67

Additional Information

**Ze-Li Dou**

Affiliation:
Department of Mathematics, Texas Christian University, P. O. Box 298900, Fort Worth, Texas 76129

Email:
z.dou@tcu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06652-2

Keywords:
Quaternionic automorphic forms,
periods of automorphic forms,
theta correspondence

Received by editor(s):
February 14, 2000

Received by editor(s) in revised form:
August 22, 2001

Published electronically:
May 22, 2002

Additional Notes:
Research for this article was partially supported by NSA Grant MDA904-97-1-0109 and by the Texas Christian University Research Fund.

Communicated by:
Dennis A. Hejhal

Article copyright:
© Copyright 2002
American Mathematical Society