On the genera of X0(N)
János A. Csirik, Joseph L. Wetherell, Michael E. Zieve
Abstract
Let g0(N) be the genus of the modular curve X0(N). We record several properties of the sequence g0(N). Even though the average size of g0(N) is 1.25N/pi2, a random positive integer has probability zero of being a value of g0(N). Also, if N is a random positive integer then g0(N) is odd with probability one. The smallest non-negative integer not occuring as g0(N) for any N is 150; the smallest such odd integer is 49267.
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