Galois theory on the line in nonzero characteristic

The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.Comment: 66 pages. Abstract added in migration

Hasse principle for generalised Kummer varieties

The existence of rational points on Kummer varieties associated to 2-coverings of abelian varieties over number fields can sometimes be proved through the variation of the Selmer group in the family of quadratic twists of the underlying abelian variety, using an idea of Swinnerton-Dyer. Following Mazur and Rubin, we consider the case when the Galois action on the 2-torsion has a large image. Under mild additional hypotheses we prove the Hasse principle for the associated Kummer varieties assuming the finiteness of relevant Shafarevich-Tate groups.Comment: 25 page

Nice equations for nice groups

Nice trinomial equations are given for unramified coverings of the affine line in nonzero characteristicp with PSL(m,q) and SL(m,q) as Galois groups. Likewise, nice trinomial equations are given for unramified coverings of the (once) punctured affine line in nonzero characteristic p with PGL(m,q) and GL(m,q) as Galois groups. Here m>1 is any integer and q>1 is any power of p

Improved rank bounds from 2-descent on hyperelliptic Jacobians

We describe a qualitative improvement to the algorithms for performing 2-descents to obtain information regarding the Mordell-Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra System and as a result, the rank bounds for hyperelliptic Jacobians are now sharper and have the conjectured parity

Second p descents on elliptic curves

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate group such that pD = C and obtains explicit models for these D as curves in projective space. This leads to a practical algorithm for performing 9-descents on elliptic curves over the rationals.Comment: 45 page