Computational Algebra Seminar: Bisson -- Computing endomorphism rings of abelian varieties

Speaker: Gaetan Bisson, Macquarie University
Title:   Computing endomorphism rings of abelian varieties
Time & Place: 3:05-4pm, Thursday 10 November, Carslaw 535

Abstract:
Jacobian varieties of hyperelliptic curves are a generalization of
elliptic curves that are just as suitable for efficient computations and
cryptographic applications. Their endomorphism ring plays a central role
in applications such as constructing varieties with a prescribed
cardinality over a prescribed finite field.

We will present the first subexponential-time algorithm for computing
the endomorphism ring structure of ordinary varieties of dimension one
and two over finite fields. It exploits the relationship between
subgroups of a variety and the ideal class group of its endomorphism
ring, which is known as complex multiplication theory.

For one-dimensional varieties, that is, elliptic curves, this algorithm
is very efficient and its complexity can be rigorously proven under just
the generalized Riemann hypothesis. In higher dimension, additional
heuristics are required, but the algorithm is nevertheless able to
perform record endomorphism ring computations.