×

Quadratic minima and modular forms. (English) Zbl 0916.11025

The main results are upper bounds on the size of the gap between the constant term and the next nonzero Fourier coefficient of an entire modular form of given weight for \(\Gamma_0(2)\), generalizing C. L. Siegel’s result [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1970, 15-56 (1970; Zbl 0225.10031)] for the full modular group. Numerical evidence indicates that a sharper bound holds for the weights \(n\equiv 2\text{ mod }4\). As an application, upper bounds are derived for the minimum positive integer represented by level-two even positive definite quadratic forms.

MSC:

11F11 Holomorphic modular forms of integral weight
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11F30 Fourier coefficients of automorphic forms

Citations:

Zbl 0225.10031
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML EMIS

References:

[1] Adelberg A., C. R. Math. Rep. Acad. Sci. Canada 14 (4) pp 173– (1992)
[2] Adelberg A., Acta Arith. 62 (4) pp 329– (1992)
[3] Adelberg A., J. Number Theory 59 (2) pp 374– (1996) · Zbl 0866.11013 · doi:10.1006/jnth.1996.0103
[4] Apostol T. M., Introduction to analytic number theory (1976) · Zbl 0335.10001
[5] Apostol T. M., Modular functions and Dirichlet series in number theory,, 2. ed. (1990) · Zbl 0697.10023
[6] Brent B., Ph.D. thesis, in: The initial segment of the Fourier series of a modular form with constant term (1994)
[7] Kimura N., Acta Arith. 50 (3) pp 243– (1988)
[8] Lehner J., Amer. J. Math. 71 pp 373– (1949) · Zbl 0032.15902 · doi:10.2307/2372252
[9] Mallows C. L., J. Algebra 36 (1) pp 68– (1975) · Zbl 0311.94002 · doi:10.1016/0021-8693(75)90155-6
[10] Miyake T., Modular forms (1989) · Zbl 0701.11014
[11] Ogg A., Modular forms and Dirichlet series (1969) · Zbl 0191.38101
[12] Schoeneberg B., Elliptic modular functions: an introduction (1974) · Zbl 0285.10016
[13] Serre J.-P., A course in arithmetic (1973) · Zbl 0256.12001
[14] Siegel C. L., Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II pp 87– (1969)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.