  We compute the number of F_q-points on M_{4,n}, for n less than or equal to
3, and show that it is a polynomial in q, using a sieve based on Hasse-Weil
zeta functions. As an application, we prove that the rational singular
cohomology groups of moduli spaces of stable curves of genus g with n marked
points vanish in all odd degrees less than or equal to 9, for all g and n. Both
results confirm predictions of the Langlands program, via the conjectural
correspondence with polarized algebraic cuspidal automorphic representations of
conductor 1, which are classified in low weight. Our vanishing result for odd
cohomology resolves a problem posed by Arbarello and Cornalba in the 1990s.
