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Proof of a conjecture of Farkas and Kra. (English) Zbl 1440.11065

Summary: In this paper we prove a conjecture of H. M. Farkas and I. Kra [Contemp. Math. 311, 115–131 (2002; Zbl 1036.30027)], which is a modular equation involving a half sum of certain modular form of weight \(1\) for congruence subgroup \(\Gamma_1(k)\) with any prime \(k\). We prove that their conjecture holds for all odd integers \(k \geq 3\). A new modular equation of Farkas and Kra type is also established.

MSC:

11F27 Theta series; Weil representation; theta correspondences
11F12 Automorphic forms, one variable
14K25 Theta functions and abelian varieties

Citations:

Zbl 1036.30027
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References:

[1] G. E. Andrews, A simple proof of Jacobi’s triple product identity, Proc. Amer. Math. Soc. 16 (1965), 333-334. · Zbl 0132.30901
[2] H. M. Farkas and I. Kra, Theta Constants, Riemann Surfaces and the Modular Group: An introduction with applications to uniformization theorems, partition identities and combinatorial number theory, Graduate Studies in Mathematics 37, American Mathematical Society, Providence, RI, 2001. · Zbl 0982.30001
[3] —-, On theta constant identities and the evaluation of trigonometric sums, in: Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), 115-131, Contemp. Math. 311, Amer. Math. Soc., Providence, RI, 2002.
[4] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Fourth edition, Cambridge University Press, New York, 1962. · Zbl 0105.26901
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