Ramin Takloo-Bighash : Rational points on one-sided compactifications of PGL(2)

Posted by Nathan Jones , part of the Number Theory Seminar.

At

Oct. 20, 2015, 11 a.m.

In

SEO 427

Abstract

In this talk I will describe some recent results about the distribution of rational points of bounded height on blowups of P^3. The varieties we consider can be thought of as compactificaions of the group PGL(2), and we will use harmonic analysis on the non-commutative group PGL(2) to obtain our results. In an old work, joint with Shalika and Tschinkel, we had studied rational points on smooth bi-equivariant compactifications of arbitrary semi-simple groups. The varieties under consideration here are only one sided compactifications. In concrete terms, this means that our height functions are not left and right invariant under the action of compact subgroups of PGL(2). This introduces delicate technical difficulties that require new ideas. This is joint work with Sho Tanimoto, and with Sho Tanimoto and Yuri Tschinkel.