Brumer's conjecture states that Stickelberger elements combining values of L-functions at s=0 for an abelian extension of number fields E/F should annihilate the ideal class group of E when it is considered as module over the appropriate group ring. In some cases, an ideal obtained from these Stickelberger elements has been shown to equal a Fitting ideal connected with the ideal class group. We consider the analog of this at s=-1, in which the class group is replaced by the tame kernel, which we will define. For a field extension of degree 2, we show that there is an exact equality etween the Fitting ideal of the tame kernel and the most natural higher Stickelberger ideal; the 2-part of this equality is conditional on the Birch-Tate conjecture.

This will be a series of two introductory lectures on the
distribution of closed points on a scheme of finite type
over the integers. Both general properties and important
examples will be discussed, with an emphasis on p-adic
variation for zeta functions over finite fields.

Motivated by the construction made by Goppa on curves, we present some error-correcting codes on algebraic surfaces. A surface whose Neron-Severi group has rank 1 has a "nice" intersection property that allows us the construction of a good code. We will verify this on specific examples. Surfaces with many points and rank 1 are not easy to find. We were able, though, to find also surfaces with low rank and many points, and these gave us good codes too. Finally, we present a decoding algorithm for these codes. It is based on the realization of the code as a LDPC code, and it is inspired on the Luby-Mitzenmacher algorithm.