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.

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.

## Date:

Mon, 06/01/2020 - 16:00 to 17:00

## Location:

Ross 63