# Mathematical Sciences Research Institute

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# Seminar

Analytic Number Theory Graduate Student Seminar March 07, 2017 (04:00 PM PST - 05:00 PM PST)
Parent Program: Analytic Number Theory MSRI: Baker Board Room
Speaker(s) Vladimir Mitankin (University of Bristol)
Description No Description
Video
Abstract/Media

4.00 pm: Carlos Pastor Alcoceba

Title:​ Lattice points in symmetric bodies.

Abstract: In a lattice point problem we are interested in estimating the number of points with integer coordinates lying inside a certain subregion of $\mathbb{R}^n$ depending on a parameter, as the parameter becomes large. Usually this family of regions correspond to dilations of a single convex body. We will see that Fourier analysis (in particular Poisson's summation formula) and exponential sums play a key role in this kind of problems. Extra symmetries can often be exploited to obtain better bounds for the error term, but technical conditions might arise.

Title: Integral points on generalised Chatelet surfaces

Abstract: For an integer a and a separable polynomial P(t) with integral coefficients we look at affine surfaces X cut out by y^2 - az^2 = P(t). Under the assumption that there is no integral Brauer-Manin obstruction and under Schinzel's hypothesis we are able to deduce that X satisfies the integral version of the Hasse principle. Furthermore, we are able to approximate in the t variable any finite collection of local points corresponding to finite primes with an integral point on X. This work is based on a classical result of Colliot-Thélène and Sansuc regarding rational points. We inject tools from algebraic number theory so that we can modify their argument in a way that works for integral points.