We present a new bound on the number of Fq-rational places in an algebraic function field. It uses information about the generators of theWeierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields. © 2008 Elsevier B.V. All rights reserved.

This is a survey paper on the distribution of algebraic points on algebraic varieties.

We show how to express a conformal map Φ of a general two connected domain in the plane such that neither boundary component is a point to a representative domain of the form Ar = {z : |z + 1/z| < 2r}, where r > 1 is a constant. The domain Ar has the virtue of having an explicit algebraic Bergman kernel function, and we shall explain why it is the best analogue of the unit disc in the two conne...

For learning about the Langlands program, knowledge of Lie-group structure theory, algebraic number theory, algebraic geometry, modular forms, and infinite-dimensional representation theory is appropriate. This article describes how one can get a glimpse of the program with less background than this.

Tannaka’s theorem states that a linear algebraic group G is determined by the category of finite-dimensional G-modules and the forgetful functor. We extend this result to linear differential algebraic groups by introducing a category corresponding to their representations and show how this category determines such a group.

Let X be a real rational surface having only weighted blowup singularities. Denote by Diffalg(X) the group of algebraic automorphisms or algebraic diffeomorphisms of X into itself. Let n be a natural integer and let e = [e1, . . . , el] be a partition of n. Denote by X e the set of l-tuples (P1, . . . , Pl) of distinct nonsingular infinitely near points of X of orders (e1, . . . , el). We show ...

We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups. This yields a structure theorem for normal equivariant embeddings of semi-abelian varieties, and a characteristic-free version of the Borel–Remmert theorem.

Let G be a simple, simply connected and connected algebraic group over an algebraically closed field of characteristic p > 0, and let V be a rational G-module such that dim V ≤ p. According to a result of Jantzen, V is completely reducible, and H(G, V ) = 0. In this paper we show that H(G, V ) = 0 unless some composition factor of V is a non-trivial Frobenius twist of the adjoint representation...

We obtain a characterization of property (T) for von Neumann algebras in terms of 1-cohomology similar to the Delorme-Guichardet Theorem for groups.

We discuss algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group-a group of twisted automor-phisms of the corresponding infinite dimensional Lie algebra. We present a complete study of dihedral reductions for sl(2, C) Lax operators with simple poles and corresponding integrable equations. In the last section we give...