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Around 1828, T. Clausen discovered that the square of certain hypergeometric F12 function can be expressed as a hypergeometric F23 function. Special cases of Clausenʼs identities were later used by S. Ramanujan in his derivation of 17 series for 1/π. Since then, there were several attempts to find new analogues of Clausenʼs identities with the hope to derive new classes of series for 1/π. Unfortunately,...
We obtain defining equations of modular curves X0(N), X1(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X0(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.
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