Number Theory Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
25 October 2018
16:00
Joni Teräväinen
Abstract


Understanding how shifts of multiplicative functions correlate with each other is a central question in multiplicative number theory. A well-known conjecture of Elliott predicts that there should be no correlation between shifted multiplicative functions unless the functions involved are ‘pretentious functions’ in a certain precise sense. The Elliott conjecture implies as a special case the famous Chowla conjecture on shifted products of the Möbius function.

In the last few years, there has been a lot of exciting progress on the Chowla and Elliott conjectures, and we give an overview of this. Nearly all of the previously obtained results have concerned correlations that are weighted logarithmically, and it is an interesting question whether one can remove these logarithmic weights. We show that one can indeed remove logarithmic averaging from the known results on the Chowla and Elliott conjectures, provided that one restricts to almost all scales in a suitable sense.

This is joint work with Terry Tao.

  • Number Theory Seminar
1 November 2018
16:00
Daniel Gulotta
Abstract

Let F be a totally real or CM number field.  Scholze has constructed Galois representations associated with torsion classes in the cohomology of locally symmetric spaces for GL_n(F).  We show that the nilpotent ideal appearing in Scholze's construction can be removed when F splits completely at the relevant prime.  As a key component of the proof, we show that the compactly supported cohomology of certain unitary and symplectic Shimura varieties with level  Gamma_1(p^{\infty}) vanishes above the middle degree. This is joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih. 

  • Number Theory Seminar
8 November 2018
16:00
Olivia Beckwith
Abstract

For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down down to -X for which the class group has trivial (non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss recent refinements of these classic results in which we consider the imaginary quadratic fields whose class number is indivisible (divisible) by p such that a given finite set of primes factor in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups satisfying almost any given finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

  • Number Theory Seminar
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