On the genera of X_0(N)

Details On the genera of X_0(N) On the genera of X_0(N)

2000-06-13

arxiv.org/abs/math/0006096v2

Mathematics

Number Theory

9 pages; minor changes in v2

Scientific paper

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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