The T-adic L-function is a unversal L-function which
interpolates classical L-functions of all p-power order
exponential sums associated to a polynomial f(x) defined
over a finite field. We study its T-adic analytic properties
(analytic continuation and its T-adic Newton polygon).
The T-adic Newton polygon provides a uniform improvement
to previous results on p-adic Newton polygon of exponential sums
in the non-ordinary case. This is joint work with Chunlei Liu.

Elliptic divisibility sequences arise as sequences of
denominators of the integer multiples of a rational point on an elliptic
curve. Silverman proved that almost every term of such a sequence has a
primitive divisor (i.e. a prime divisor that has not appeared as a
divisor of earlier terms in the sequence). If the elliptic curve has
complex multiplication, then we show how the endomorphism ring can be
used to index a similar sequence and we prove that this sequence also
has primitive divisors. The original proof fails in this context and
will be replaced by an inclusion-exclusion argument and sharper
diophantine estimates.

I will discuss some results on the vanishing and non-vanishing of
critical values of L-functions and their derivatives, both experimental and
theoretical. I will present an example of a computational "elliptic Stark
point" in a cyclic quintic extension of the rationals.

Consider a rational map φ on the projective line, from which we form a (discrete) dynamical system via iteration, and let K be a number field. A fundamental question in arithmetic dynamics is the uniform boundedness conjecture of Morton and Silverman, which states that there is a constant independent of φ (depending only on its degree) giving an upper bound for the number of K-rational preperiodic points of φ. This is a deep conjecture, and no specific case of it is known. I have proposed a specific version of the conjecture: that in the case of a degree-2 rational map and K = Q, the upper bound is 12.

In this talk, which assumes no previous knowledge of arithmetic dynamics, I will describe why this question is so difficult and sketch work that has been done to date, including giving justification for my refined uniform boundedness conjecture. The techniques used so far, which have clear limitations, involve constructing algebraic curves parameterizing maps $\phi$ together with points of period n for varying n (so-called dynamic modular curves).