preprint, 2000.

(Prior to publication, this paper should be cited as arXiv:math/0006096.)

Let *X*_{0}(*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: lim_{B→∞}(1/*B*)∑_{N≤B} *g*(*N*)/*N* = 1.25/π^{2}.

(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 *X*_{0}(*N*) is defined over
*k*=**F**_{p2}, 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 *X*_{0}(*N*)
is the dimension of the space of weight-2 cuspidal modular forms on
*Γ*_{0}(*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*:
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