Abstract. In this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.
Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of M_{4}.
In Paper II we consider the moduli space Hg,n of n-pointed smooth hyperelliptic curves of genus g. We find that there are recursion formulas in the genus that the numbers of points of 'H_{g,n} fulfill. Thus, if we can make \mathbb{S}_{n}-equivariant counts of "H_{g,n} for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to M_{2,n} for n up to seven, and give us the \mathbb{S}_{n}-equivariant Hodge structure of their cohomology. Moreover, we find that the \mathbb{S}_{n}-equivariant counts of H_{g,n} depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.
In Paper III we consider the moduli space Q_{n} of smooth n-pointed non-hyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make Sn-equivariant counts of the numbers of points of Q_{n} for n up to seven. We also count pointed plane cubics. This gives us \mathbb{S}_{n}-equivariant counts of the moduli space M_{1,n} for n up to ten. We can then determine the \mathbb{S}_{n}-equivariant Hodge structure of the cohomology of M_{3,n} for n up to five.