New results on automorphic integrals and their period functions

Author:
Richard A. Cavaliere

Journal:
Trans. Amer. Math. Soc. **301** (1987), 401-412

MSC:
Primary 11F03

MathSciNet review:
879581

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Automorphic integrals, being generalizations of automorphic forms on discrete subgroups of , share properties similar to those of forms. In this article I obtain a natural boundary result for integrals which is similar to that which holds for forms. If an automorphic integral on a given group behaves like a form on a subgroup of finite index (i.e., the period functions are identically zero), then in fact the integral must be a form on the whole group. Specializing to modular integrals with integer dimension I obtain a lower bound on the number of poles of the period functions which, of necessity, lie in quadratic extensions of the rationals.

**[1]**R. A. Cavaliere,*Automorphic integrals and their period functions*, Doctoral Dissertation, Temple University, 1984.**[2]**Marvin I. Knopp,*Some new results on the Eichler cohomology of automorphic forms*, Bull. Amer. Math. Soc.**80**(1974), 607–632. MR**0344454**, 10.1090/S0002-9904-1974-13520-2**[3]**Marvin I. Knopp,*Rational period functions of the modular group*, Duke Math. J.**45**(1978), no. 1, 47–62. With an appendix by Georges Grinstein. MR**0485700****[4]**Marvin I. Knopp,*Rational period functions of the modular group. II*, Glasgow Math. J.**22**(1981), no. 2, 185–197. MR**623004**, 10.1017/S0017089500004663**[5]**Joseph Lehner,*Discontinuous groups and automorphic functions*, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR**0164033****[6]**Joseph Lehner,*A short course in automorphic functions*, Holt, Rinehart and Winston, New York-Toronto, Ont.-London, 1966. MR**0201637****[7]**Hans Petersson,*Theorie der automorphen Formen beliebiger reeller Dimension und ihre Darstellung durch eine neue Art Poincaréscher Reihen*, Math. Ann.**103**(1930), no. 1, 369–436 (German). MR**1512629**, 10.1007/BF01455702

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
11F03

Retrieve articles in all journals with MSC: 11F03

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0879581-8

Article copyright:
© Copyright 1987
American Mathematical Society