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On the genera of X_0(N)

János A. Csirik·Joseph L. Wetherell·Michael E. Zieve
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Abstract

Let g_0(N) be the genus of the modular curve X_0(N). We record several properties of the sequence {g_0(N)}. Even though the average size of g_0(N) is 1.25N/pi^2, a random positive integer has probability zero of being a value of g_0(N). Also, if N is a random positive integer then g_0(N) is odd with probability one. The smallest non-negative integer not occuring as g_0(N) for any N is 150; the smallest such odd integer is 49267.

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János A. Csirik, Joseph L. Wetherell, Michael E. Zieve. 2000-09-05. On the genera of X_0(N). https://arxiv.org/abs/math/0006096

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