On evaluations of the modular $j$--invariant by modular equations of degree 2

Dae Hyun Paek; Jinhee Yi

doi:10.4134/BKMS.2015.52.1.263

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Bull. Korean Math. Soc. 2015; 52(1): 263-273

Printed January 31, 2015

https://doi.org/10.4134/BKMS.2015.52.1.263

On evaluations of the modular $j$--invariant by modular equations of degree 2

Dae Hyun Paek and Jinhee Yi

Busan National University of Education, Korea Science Academy of KAIST

Abstract

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Abstract

We derive modular equations of degree 2 to establish explicit relations for the parameterizations for the theta functions $\varphi$ and $\psi$. We then find specific values of the parameterizations to evaluate some new values of the modular $j$--invariant in terms of $J_n$.

Keywords: theta functions, modular equations, modular $j$--invariant

MSC numbers: Primary 11F27, 11R29, 33C90; Secondary 33C05, 33E05