(Prior to publication, this paper should be cited as arXiv:math/0006096.)
Let X0(N) be the
classical
modular curve of level N, and let
g(N) be its genus. We prove the following results
about g(N):
(1): Upper and lower bounds, including the asymptotic results
0 < lim inf g(N)/N < lim sup g(N)/(N log log N) < ∞.
(2): Average behavior: limB→∞(1/B)∑N≤B g(N)/N = 1.25/π2.
(3): Natural density: {g(N)} is a density zero subset of the integers.
(4): Non-uniformity of g(N) modulo primes:
for instance, g(N) is odd with probability 1, and (for a fixed odd
prime p) the probability that g(N) ≡ 1 (mod p)
is much less than 1/p.
Properties (2) and (3) imply there is much collapsing under the map N→g(N); for instance, there are integers whose preimage under this map is arbitrarily large. Also, we note that data for small N is misleading: the smallest odd positive integer which does not occur as a value of g(N) is 49267, but still the density of odd integers occuring as such values is zero.
Our study of g(N) was motivated by a 1983 question of Serre, which asked whether (over a fixed finite field k) there exist curves of every genus g which have O(g) rational points. Since every supersingular point on X0(N) is defined over k=Fp2, this curve has O(g) points over k, so one might hope that its genus achieves all or nearly all values. However, our results show that this is not the case. We have answered Serre's question via a different argument in a subsequent paper.
Additional comment from May 2005: The genus of X0(N) is the dimension of the space of weight-2 cuspidal modular forms on Γ0(N); Greg Martin has generalized our upper and lower bounds, and our average value result, to the dimensions of the spaces of cusp forms of arbitrary weight.
Michael Zieve: home page publication list