I will explain how to produce, for any given number N, an elliptic curve over a finite field having exactly N points. If N is prime, the heuristic run time is polynomial in log(N).

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.