## Taiwanese Journal of Mathematics

### Proof of a Conjecture of Farkas and Kra

Nian Hong Zhou

#### Abstract

In this paper we prove a conjecture of Farkas and Kra, 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.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 6 (2019), 1317-1326.

Dates
Revised: 23 January 2019
Accepted: 10 March 2019
First available in Project Euclid: 13 March 2019

https://projecteuclid.org/euclid.twjm/1552442420

Digital Object Identifier
doi:10.11650/tjm/190301

Mathematical Reviews number (MathSciNet)
MR4033547

Zentralblatt MATH identifier
07142975

#### Citation

Zhou, Nian Hong. Proof of a Conjecture of Farkas and Kra. Taiwanese J. Math. 23 (2019), no. 6, 1317--1326. doi:10.11650/tjm/190301. https://projecteuclid.org/euclid.twjm/1552442420

#### References

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• ––––, 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.
• 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.