Galois groups of finite extensions of number fields naturally act on many finitely generated abelian groups, such as on the ring of integers of the number field, or on the units therein, or on Mordell-Weil groups of elliptic curves defined over the base field. Quotioning out the torsion gives an integral Galois representation, i.e. a homomorphism from the Galois group to GL_n(Z), and it is a classical problem, going back to Emmy Noether, to understand the structure of such representations. Since for most finite groups, there is no classification of integral representations, it is not even clear a priori what a satisfactory answer might look like. But as it turns out, and as I shall explain in my talk, there is a mysterious connection between the structure of these Galois modules on the one hand, and other classical number theoretic invariants, such as class groups of number fields, or Tate-Shafarevich groups of elliptic curves on the other hand. Part of the work to be presented is joint with Bart de Smit.
Let p be a prime. Let H denote the maximal unramified abelian extension of K:=Q(pth roots of unity) such that [H:K] is prime to p (so H is the Hilbert class field if p is regular). Call p abelian if p=3(mod 4) and the extension H/ Q(sqrt(-p)) is abelian. One can check that 23 and 31 are the first two abelian primes. I will show that the set of abelian primes is finite. I will discuss the proof of this result, which relies on Serre's conjecture, as well as generalizations to general cyclotomic fields. I may also speak about how (not fully proven) generalizations of Serre's conjecture may need to analogous results.
Let G be a finite group and d(G) the minimal number of conjugacy classes that generate G. In any tame realization of G as a Galois group over Q there are at least d(G) ramified primes. The (tame) minimal ramification problem asks whether any group G can be realized (tamely) over Q with exactly d(G) ramified primes. We prove that this problem has an affirmative answer for a substantial class of finite nilpotent groups. (Joint work with Hershy Kisilevsky and Jack Sonn)
Suppose we are given a smooth, proper, geometrically connected curve X in characteristic p with an action of a finite group G. Does there exist a smooth, proper curve X' with G-action in characteristic zero such that X' (with G-action) lifts X (with G-action)? It turns out that solving this lifting problem reduces to solving a local lifting problem in a formal neighborhood of each point of X where G acts with non-trivial inertia. The Oort conjecture states that this local lifting problem should be solvable whenever the inertia group is cyclic. A new result of Stefan Wewers and the speaker shows that the local lifting problem is solvable whenever the inertia group is cyclic of order not divisible by p^4, and in many cases even when the inertia group is cyclic and arbitrarily large. We will discuss this result, after giving some background on the local lifting problem in general.
Last modification: 22 February 2012.