Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level 2 structure, together with a computation of Euler characteristics, we find the isotypical decomposition, under the symmetric group on six letters, of spaces of vector-valued Siegel modular forms of degree 2 and level 2.

This paper goes beyond Katz–Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally, and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus g≥3. The experiments point to a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus ≥3. However, for elliptic curves and for hyperelliptic curves of every genus, we prove that this stronger convergence cannot occur.

We compute the stable cohomology of moduli spaces of hyperelliptic curves of a fixed genus embedded in a fixed Hirzebruch surface. We also describe these moduli spaces of embedded hyperelliptic curves in terms of moduli spaces of pointed non-embedded hyperelliptic curves.

We compute the number of F-q-points on (M) over bar (4,n) for n <= 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 group H-k((M) over bar (g,n)) vanishes for all odd k <= 9. 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.

We use a classification result of Chenevier and Lannes for algebraic automorphic representations together with a conjectural correspondence with ℓ-adic absolute Galois representations to determine the Euler characteristics (with values in the Grothendieck group of such representations) of and for n ≤ 14 and of local systems on for |λ| ≤ 16.

For a given genus g ≥ 1, we give lower bounds for the maximal number of rational pointson a smooth projective absolutely irreducible curve of genus g over F_q. As a consequenceof Katz–Sarnak theory, we first get for any given g > 0, any ε > 0 and all q large enough,the existence of a curve of genus g over F_q with at least 1 + q + (2g − ε)√q rational points.Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lowerbound of the form 1 + q + 1.71√q valid for g ≥ 3 and odd q ≥ 11. Finally, explicit constructions of towers of curves improve this result: We show that the bound 1 + q + 4√q − 32 isvalid for all g ≥ 2 and for all q.

We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical lifting to characteristic zero, that is, a lifting for which the reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between abelian varieties over finite fields and fractional ideals in étale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations.

We provide explicit generators of the torsion of the second cohomology of bielliptic surfaces, and we use this to study the pullback map between the Brauer group of a bielliptic surface and that of its canonical cover.

We formulate a detailed conjectural Eichler–Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes GU(2, 1,  Q (√-3)). The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula.

Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems.

Let q be a power of a prime number p, F-q be a finite field with q elements and G be a subgroup of (F-q,+) of order p. We give an existence criterion and an algorithm for computing maximally G-fixed c-Wieferich primes in F-q[T]. Using the criterion, we study how c-Wieferich primes behave in F-q[T] extensions.