NTS ABSTRACT: Difference between revisions
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{ style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
    
 bgcolor="#F0A0A0" align="center" style="fontsize:125%"    bgcolor="#F0A0A0" align="center" style="fontsize:125%"  Frank Thorne  
    
 bgcolor="#BCD2EE" align="center"    bgcolor="#BCD2EE" align="center"  Levels of distribution for prehomogeneous vector spaces  
    
 bgcolor="#BCD2EE"    bgcolor="#BCD2EE"  One important technical ingredient in many arithmetic statistics papers is  
upper bounds for finite exponential sums which arise as Fourier transforms  
of characteristic functions of orbits. This is typical in results  
obtaining power saving error terms, treating "local conditions", and/or  
applying any sort of sieve.  
In my talk I will explain what these exponential sums are, how they arise,  
and what their relevance is. I will outline a new method for explicitly and easily  
evaluating them, and describe some pleasant surprises in our end results. I will also  
outline a new sieve method for efficiently exploiting these results, involving  
Poisson summation and the BhargavaEkedahl geometric sieve. For example, we have proved  
that there are "many" quartic field discriminants with at most eight  
prime factors.  
This is joint work with Takashi Taniguchi.  
Revision as of 16:54, 23 January 2017
Return to NTS Spring 2017
Jan 19
Bianca Viray 
On the dependence of the BrauerManin obstruction on the degree of a variety 
Let X be a smooth projective variety of degree d over a number field k. In 1970 Manin observed that elements of the Brauer group of X can obstruct the existence of a kpoint, even when X is everywhere locally soluble. In joint work with Brendan Creutz, we prove that if X is geometrically abelian, Kummer, or bielliptic then this BrauerManin obstruction to the existence of a kpoint can be detected from only the dprimary torsion Brauer classes. 
Jan 26
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Feb 2
Arul Shankar 
Bounds on the 2torsion in the class groups of number fields 
(Joint with M. Bhargava, T. Taniguchi, F. Thorne, J. Tsimerman, Y. Zhao)
Given a number field K of fixed degree n over Q, a classical theorem of BrauerSiegel asserts that the size of the class group of K is bounded by O_\epsilon(Disc(K)^(1/2+\epsilon). For any prime p, it is conjectured that the ptorsion subgroup of the class group of K is bounded by O_\epsilon(Disc(K)^\epsilon. Only the case n=p=2 of this conjecture in known. In fact, for most pairs (n,p), the best known bounds come from the "convex" BrauerSiegel bound. In this talk, we will discuss a proof of a subconvex bound on the size of the 2torsion in the class groups of number fields, for all degrees n. We will also discuss an application of this result towards improved bounds on the number of integral points on elliptic curves. 
Feb 9
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Feb 16
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Feb 23
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Mar 2
Frank Thorne 
Levels of distribution for prehomogeneous vector spaces 
One important technical ingredient in many arithmetic statistics papers is
upper bounds for finite exponential sums which arise as Fourier transforms of characteristic functions of orbits. This is typical in results obtaining power saving error terms, treating "local conditions", and/or applying any sort of sieve. In my talk I will explain what these exponential sums are, how they arise, and what their relevance is. I will outline a new method for explicitly and easily evaluating them, and describe some pleasant surprises in our end results. I will also outline a new sieve method for efficiently exploiting these results, involving Poisson summation and the BhargavaEkedahl geometric sieve. For example, we have proved that there are "many" quartic field discriminants with at most eight prime factors. This is joint work with Takashi Taniguchi.

Mar 9
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Mar 16
Mar 30
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Apr 6
Celine Maistret 
Apr 13
Apr 20
Yueke Hu 
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Apr 27
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May 4
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