Computational Algebra Seminar: Frengley -- Minimisation algorithms over function fields and applications

Speaker: Sam Frengley (Bristol)
Title: Minimisation algorithms over function fields and applications
Time & Place: 15:00-1600, Thursday 30 October, SMRI Seminar Rm
Abstract:
"Minimisation" algorithms (and their sibling "reduction"
algorithms) have proved a very fruitful tool in number theory, dating at
least to Gauss' study of integral binary quadratic forms. Since then,
these algorithms have seen a remarkable variety of applications in
number theory. In computation they are regularly used to study (for
example) class groups, descent on elliptic curves, reduction of
quadratic forms. On the other hand, they also play a pivotal role in
many theoretical results, notably in great advances in arithmetic
statistics in recent decades.

I will discuss the (folklore) "geometric" versions of these algorithms,
exploiting the analogy between number fields and function fields (or
spectra of rings of integers and curves). I will illustrate their
utility using some examples arising from Hilbert modular surfaces
leading to minimising conics over QQ(x,y) (joint with Alex Cowan and
Kimball Martin) and small degree covers of PP^2. Time permitting, I may
discuss a more theoretical application: classifying which rational
scrolls contain degree 5 covers of PP^1 (a case of the Tschirnhausen
realisation problem) which is joint with Sameera Vemulapalli.