Let $C$ be a curve over a local field whose reduction is totally
degenerate. We discuss the related problems of 1) determining the
group structure of the torsion subgroup of the Jacobian of $C$, and 2)
determining if a given line bundle on $C$ is divisible by a given
integer $r$. Under certain hypotheses on the reduction of $C$, we
exhibit explicit algorithms for answering these two questions.

Every finite field has many multiplicative generators. However,
finding one in polynomial time is an important open problem .
In fact, even finding elements of high order has not been solved
satisfactorily. In this paper, we present an algorithm that for any
positive integer $c$ and prime power $q$, finding an element of
order $\exp(\Omega(\sqrt{q^c}) ) $ in the finite field $\F_{q^{(q^c-1)/(q-1)}}$
in deterministic time $(q^c)^{O(1)}$. We also show that there are
$\exp(\Omega(\sqrt{q^c}) ) $ many weak keys for the discrete logarithm problems
in those fields with respect to certain bases.

Serre proved in 1972 that the image of the adelic Galois representation
associated to an elliptic curve E without complex multiplication has open
image; moreover, he also proved that for an elliptic curve over Q the
index of the image is always divisible by 2 (and in particular never
surjective). More recently, Greicius in his thesis gave criteria for
surjectivity and gave an explicit example of an elliptic curve E over a
number field K with surjective adelic representation. Soon after, Zywina,
building on earlier work of Duke, Jones, and others, proved that the
adelic image `random' elliptic curve is maximal.

In this talk I will explain recent joint work with David Zywina in which
we generalize these theorems and prove that a random abelian variety in a
family with big monodromy has maximal image of Galois. I'll explain what
big monodromy and maximal mean an explain the analytic and geometric
techniques used in previous work and the new geometric ideas -- in
particular, Nori's method of semistable approximation-- needed to
generalized to higher dimension.

For an integer n > 2, the unit group modulo n has an even number
of elements, with half of them having representatives in (0,n/2)
and the other half having representatives in (n/2,n). It is
"balanced". Say a subgroup H of this unit group is "balanced"
if each coset of H is evenly split between the bottom half and
the top half. Suppose g>1 is a fixed integer. We are concerned with
the distribution of numbers n coprime to g for which the
cyclic subgroup in the unit group mod n is balanced.
This has an application to the statistical study of the rank
of the Legendre curve over function fields. (Joint work with
Douglas Ulmer.)

Among the most important complexity problems in
coding theory are the maximun likelihood decoding
and the computation of the minimun distance.
In this talk, we explain a self-contained, quick and
transparent proof of the NP-hardness of these
problems based on the subset sum problem over
finite fields.