Let N(f) denote the number of zeros of a sparse
multivariate polynomial f(x) over a finite field of
characteristic p. In this lecture, we discuss the
complexity and algorithms for computing the
reduction N(f) modulo a power of p.

Let G be a finite abelian group. A zero-sum problem on G asks for
the smallest positive integer k such that for any sequence a_1,...,a_k
of elements of G there exists a subsequence of required length the sum of
whose terms vanishes. In this talk we will give a survey of problems and
results in this field. In particular, we will talk about Olson's theorem
on the Davenport constanst of an abelian p-group and Reiher's celebrated
proof of the Kemnitz conjecture.

The set of multiplicative arithmetic functions over a ring R
(commutative with identity) can be given a unique functorial ring
structure for which (1) the operation of addition is Dirichlet
convolution and (2) multiplication of completely multiplicative
functions coincides with point-wise multiplication. This existence of
this ring structure can be derived from the existence of the ring of
``big'' Witt vectors, and it yields a ring structure on the set of
formal Dirichlet series that are expressible as an Euler product. The
group of additive arithmetic functions over R also has a naturally
defined ring structure, and there is a functorial ring homomorphism
from the ring of multiplicative functions to the ring of additive
functions that is an isomorphism if R is a Q-algebra. An application
is given to zeta functions of schemes of finite type over the ring
of integers.

Varieties with an action of a reductive group such that the Borel subgroup has an open orbit are called spherical. Spherical varieties are a unifying theme behind many analytic techniques in the theory of automorphic forms, such as the relative trace formula and integral representations of L-functions. After briefly surveying these methods - for which a general and systematic theory is missing - in order to justify this claim, I prove a general result on the representation theory of spherical varieties for split groups over p-adic fields: Irreducible quotients of the "unramified summand" of Cc&#8734(X) (where X is the spherical variety) are "roughly" parametrized by the quotient of a complex torus by a finite reflection group. This generalizes the classical parametrization of the "unramified spectrum" of G by semisimple conjugacy classes in its Langlands dual, and is compatible with recent results of D.Gaitsgory & D.Nadler which assign a "Langlands dual group" to every spherical variety. The main tool in the proof is an action, defined by F.Knop, of the Weyl group of G on the set of Borel orbits on X.