A heuristic for the distribution of point counts for random curves over a finite field.

How many rational points are there on a random algebraic curve of large genus g over a given finite field Fq? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q+1+1/(q-1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g.

π 1 of Miranda moduli spaces of elliptic surfaces.

We give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over P 1 which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of S L 2 Z presented in terms of ( 1 1 0 1 ) , ( 1 0 - 1 1 ) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other. Our approach exploits the natural Z 2 -action on Weierstrass curves and the identification of Z 2 -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over P 1 . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition.

Operations on stable moduli spaces.

We construct certain operations on stable moduli spaces and use them to compare cohomology of moduli spaces of closed manifolds with tangential structure. We obtain isomorphisms in a stable range provided the p-adic valuation of the Euler characteristics agree, for all primes p not invertible in the coefficients for cohomology.