Computational Algebra Seminar: Kedlaya - Brown

NOTE UNUSUAL DAY!
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Speaker: Kiran Kedlaya (MIT)
Title: Computing L-series for low genus curves
Time & Place: 3:00-3:50pm, Tuesday 19 Jan, Carslaw 535

Abstract: Joint work with Andrew Sutherland (MIT).
We make a practical comparison of several algorithms for computing
the zeta functions of curves of genus up to 3 over finite fields,
and computing L-series of such curves over number fields. These include
optimized enumeration of points, generic group algorithms improving
upon the Shanks baby step-giant step approach, and p-adic cohomology
(work of David Harvey). If time permits, I may also touch upon some
observations regarding exceptional distributions of Frobenius
eigenvalues (i.e., higher-genus analogues of the CM exception to
the Sato-Tate law).

Speaker: Gavin Brown (Loughborough)
Title: Rational points on 3-folds
Time & Place: 4:10-5:00pm, Tuesday 19 Jan, Carslaw 535

Abstract: Bogomolov and Tschinkel showed that many nonsingular
hypersurfaces of low degree and low dimension are densely covered
by rational points (after allowing a finite extension of the base field).
One of their methods involved building elliptic fibrations with good
properties. I will sketch the ideas behind this and then show how to
adapt methods of Cheltsov and Park to show the same in other concrete
contexts. To some extent, this is about taking an ideal and working out
how to push some of its variables into the coefficient field so that the
rest resemble the ideal of an elliptic curve; sometimes you have to
add new variables in a careful way to make it work, which I will
illustrate with examples.