×

A test for identifying Fourier coefficients of automorphic forms and application to Kloosterman sums. (English) Zbl 0966.11018

The author describes a numerical test in order to determine whether a given sequence of numbers can occur as Fourier coefficients of a Maass wave form, without knowing its eigenvalue. The test can be extended to holomorphic newforms as well. In particular a negative answer is obtained for the Kloosterman sums \(\pm S(1,1;p)/ \sqrt p\) for primes \(p\).
Reviewer: A.Krieg (Aachen)

MSC:

11F30 Fourier coefficients of automorphic forms
11L05 Gauss and Kloosterman sums; generalizations
11Y99 Computational number theory
PDF BibTeX XML Cite
Full Text: DOI Euclid EuDML

References:

[1] Brumer A., Columbia University Number Theory Seminar (New York, 1992) pp 41– (1995)
[2] DOI: 10.1017/CBO9780511609572
[3] Casselman W., Algebraic number fields: L-functions and Galois properties (Durham, 1975) pp 663– (1977)
[4] Chai C., ”Character sums and automorphic forms” (1999)
[5] Cohen H., SĂ©minaire de Theorie des Nombres de Bordeaux (1977)
[6] Hejhal D. A., Math. Comp. 61 (203) pp 245– (1993)
[7] Katz N., Sommes exponentielles (1980)
[8] Knapp A., Elliptic curves (1992) · Zbl 0804.14013
[9] DOI: 10.1007/BF01329622
[10] DOI: 10.1007/978-1-4612-4816-3_19
[11] DOI: 10.1016/0097-3165(91)90016-A · Zbl 0729.11065
[12] Serre J.-P., Abelian l-adic representations and elliptic curves (1968)
[13] DOI: 10.1090/S0894-0347-97-00220-8 · Zbl 0871.11032
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.