Dirichlet series and automorphic forms on unitary groups

Author:
Tobias Orloff

Journal:
Trans. Amer. Math. Soc. **290** (1985), 431-456

MSC:
Primary 11F55

DOI:
https://doi.org/10.1090/S0002-9947-1985-0792806-0

MathSciNet review:
792806

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Abstract: In a special case our unitary group takes the form

Here

is a skew-Hermitian matrix with entries in an imaginary quadratic number field

. We suppose that

has signature

. This group acts naturally on the symmetric domain

If

with

the ring of integers in

, then an automorphic form

with respect to

has an expansion

. The functions

are theta functions. Given another automorphic form

with an expansion

we define a Dirichlet series

. Here

is a certain positive definite inner product on the space of theta functions. The series is obtained as an integral of Rankin type:

with

a subgroup of "translations". The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over

. In the case

our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type.

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S. Kudla, *On certain Euler products for* , Compositio Math. **42** (1981), 321-344. MR **607374 (83d:10032)**
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S. Gelbart and I. I. Piatetski-Shapiro, *Automorphic forms and* -*functions for the unitary group*, preprint 1983.
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I. I. Piatetski-Shapiro, *Tate theory for reductive groups and distinguished representations*, Proc. Internat. Congress Math. (Helsinki, 1978), Acad. Sci. Fenn, 1980, pp. 585-590. MR **562659 (82f:10037)**

**[1]**- A. Borel,
*Reduction theory for arithmetic groups*, Algebraic Groups and Discontinuous Subgroups, Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, R. I., 1966. MR **0204533 (34:4372)**
**[2]**- Y. Flicker, -
*packets and liftings for* , preprint.
**[3]**- R. P. Langlands,
*On the functional equations satisfied by Eisenstein series*, Lecture Notes in Math., vol. 544, Springer, Berlin and New York, 1976. MR **0579181 (58:28319)**
**[4]**- O. T. O'Meara,
*Introduction to quadratic forms*, Springer-Verlag, Berlin and New York, 1971. MR **0347768 (50:269)**
**[5]**- G. Shimura,
*Arithmetic of unitary groups*, Ann. of Math. **79** (1964), 369-409. MR **0158882 (28:2104)**
**[6]**- -,
*On modular forms of half integral weight*, Ann. of Math. **97** (1973), 440-481. MR **0332663 (48:10989)**
**[7]**- -,
*The arithmetic of automorphic forms with respect to a unitary group*, Ann. of Math. **107** (1978), 569-605. MR **0563087 (58:27797)**
**[8]**- -,
*The special values of zeta functions associated with Hilbert modular forms*, Duke Math. J. **45** (1978), 637-679. MR **507462 (80a:10043)**
**[9]**- -,
*The arithmetic of certain zeta functions and automorphic forms on orthogonal groups*, Ann. of Math. **111** (1980), 313-375. MR **569074 (81g:10041)**
**[10]**- T. Shintani,
*On automorphic forms on unitary groups of order* , preprint.
**[11]**- S. Kudla,
*On certain Euler products for* , Compositio Math. **42** (1981), 321-344. MR **607374 (83d:10032)**
**[12]**- S. Gelbart and I. I. Piatetski-Shapiro,
*Automorphic forms and* -*functions for the unitary group*, preprint 1983.
**[13]**- I. I. Piatetski-Shapiro,
*Tate theory for reductive groups and distinguished representations*, Proc. Internat. Congress Math. (Helsinki, 1978), Acad. Sci. Fenn, 1980, pp. 585-590. MR **562659 (82f:10037)**

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0792806-0

Article copyright:
© Copyright 1985
American Mathematical Society