We sharpen our previous results on the g and n such that M g,n is of general type for some g with g + 1 prime.

We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in MAGMA. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of III [2], for families of curves of genus 2 of the form y2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.

Abstract. We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.

, d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in ‫ސ‬ 3 defined by ax= 0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic topology.

Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relation in a general geometric context. We also prove the relation for one example of a modular form of weight 4 and two of weight 3, refine the statement and suggest a method of proving it for four more of weight 4, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).