## 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

### Keywords:

theta functions; theta constants; modular equations

Zbl 1036.30027
Full Text:

### 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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