# Past Junior Number Theory Seminar

3 March 2014
16:00
Christophe Petit
Abstract
The elliptic curve discrete logarithm problem (ECDLP) is commonly believed to be much harder than its finite field counterpart, resulting in smaller cryptography key sizes. In this talk, we review recent results suggesting that ECDLP is not as hard as previously expected in the case of composite fields. We first recall how Semaev's summation polynomials can be used to build index calculus algorithms for elliptic curves over composite fields. These ideas due to Pierrick Gaudry and Claus Diem reduce ECDLP over composite fields to the resolution of polynomial systems of equations over the base field. We then argue that the particular structure of these systems makes them much easier to solve than generic systems of equations. In fact, the systems involved here can be seen as natural extensions of the well-known HFE systems, and many theoretical arguments and experimental results from HFE literature can be generalized to these systems as well. Finally, we consider the application of this heuristic analysis to a particular ECDLP index calculus algorithm due to Claus Diem. As a main consequence, we provide evidence that ECDLP can be solved in heuristic subexponential time over composite fields. We conclude the talk with concrete complexity estimates for binary curves and perspectives for furture works. The talk is based on joint works with Jean-Charles Faugère, Timothy Hodges, Yung-Ju Huang, Ludovic Perret, Jean-Jacques Quisquater, Guénaël Renault, Jacob Schlatter, Naoyuki Shinohara, Tsuyoshi Takagi
• Junior Number Theory Seminar
17 February 2014
16:00
Benjamin Green
Abstract

In this talk I will explain the basic motivation behind the trace formula and give some simple examples. I will then discuss how it can be used to prove things about automorphic representations on general reductive groups.

• Junior Number Theory Seminar
10 February 2014
16:00
Abstract

A finite set of elements in a connected real Lie group is "Diophantine" if non-identity short words in the set all lie far away from the identity. It has long been understood that in abelian groups, such sets are abundant. In this talk I will discuss recent work of Aka; Breuillard; Rosenzweig and de Saxce concerning this phenomenon (and its limitations) in the more general setting of nilpotent groups.

• Junior Number Theory Seminar
3 February 2014
16:00
Simon Myerson
Abstract
<p><span>In 1997 V.&nbsp;</span><span>Bentkus and F. G</span><span>ötze introduced a technique for estimating $L^p$ norms of certain exponential sums without needing an explicit estimate for the exponential sum itself. One uses instead a kind of estimate I call a "moat lemma". I explain this term, and discuss the implications for several kinds of point-counting problem which we all know and love.</span></p>
• Junior Number Theory Seminar
27 January 2014
16:00
Eugen Keil
Abstract
In the first part of the talk we are going to build up some intuition about limit-periodic functions and I will explain why they are the 'simplest' class of arithmetic functions appearing in analytic number theory. In the second part, I will give an equivalent description of 'limit-periodicity' by using exponential sums and explain how this property allows us to solve 'twin-prime'-like problems by the circle method.
• Junior Number Theory Seminar
20 January 2014
16:00
Abstract
<p>This talk will discuss the discovery of Heegner points from a historic perspective. They are a beautiful application of analytic techniques to the study of rational points on elliptic curves, which is now a ubiquitous theme in number theory. We will start with a historical account of elliptic curves in the 60's and 70's, and a correspondence between Birch and Gross, culminating in the Gross-Zagier formula in the 80's. Time permitting, we will discuss certain applications and ramifications of these ideas in modern number theory.&nbsp;</p>
• Junior Number Theory Seminar
2 December 2013
17:00
Freddie Manners
Abstract
The 'pyjama stripe' is the subset of the plane consisting of a vertical strip of width epsilon about every integer x-coordinate. The 'pyjama problem' asks whether finitely many rotations of the pyjama stripe about the origin can cover the plane. I'll attempt to outline a solution to this problem. Although not a lot of this is particularly representative of techniques frequently used in additive combinatorics, I'll try to flag up whenever this happens -- in particular ideas about 'limit objects'.
• Junior Number Theory Seminar
25 November 2013
17:00
Francesca Balestrieri
Abstract
This talk will be a gentle introduction to the main ideas behind some of the obstructions to the Hasse principle. In particular, I'll focus on the Brauer-Manin obstruction and on the descent obstruction, and explain briefly how other types of obstructions could be constructed.
• Junior Number Theory Seminar
18 November 2013
17:00
Jan Dumke
Abstract
In the 1930's E. Artin conjectured that a form over a p-adic field of degree d has a non-trivial zero whenever n>d^2. In this talk we will discuss this relatively old conjecture, focusing on recent developments concerning quartic and quintic forms.
• Junior Number Theory Seminar