Title: Cup products oncurves over finite fields
Abstract: This is joint work with Ted Chinburg.
Let C be a smooth projective curve over a finite field k, and
let l be a prime number different from the characteristic of k.
In this talk I will discuss triple cup products on the first etale
cohomology group of C with coefficients in the constant
sheaf of l-th roots of unity. These cup products are important
for finding explicit descriptions of the l-adic completion of the
etale fundamental group of C and also for cryptographic
applications. For the latter, one restricts the cup product to
triples of cyclic groups of order l inside the first cohomology
group. I will describe an upper and a lower bound for the
number of different non-degenerate trilinear maps obtained
this way. Using work of McCallum and Sharifi on cup products
in Iwasawa theory, I will also present a formula of the value
of the triple cup product on a given element in its domain. We
do not know if this formula can in general be computed in
polynomial time.