Forthcoming events in this series


Mon, 25 Nov 2013

17:00 - 18:00
C5

Obstructions to the Hasse principle

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.

Mon, 18 Nov 2013

17:00 - 18:00
C5

Artin's conjecture on p-adic forms

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.

Mon, 11 Nov 2013

17:00 - 18:00
C5

Cubic polynomials represented by norm forms

Alastair Irving
Abstract

I will describe how a sieve method can be used to establish the Hasse principle for the variety

$$f(t)=N(x_1,\ldots,x_k),$$

where $f$ is an irreducible cubic and $N$ is a norm form for a number field satisfying certain hypotheses.

Mon, 28 Oct 2013

17:00 - 18:00
C5

Mixed Motives in Number Theory

Netan Dogra
Abstract

Mixed motives turn up in number theory in various guises. Rather than discuss the rather deep foundational questions involved, this talk will aim

to give several illustrations of the ubiquity of mixed motives and their realizations. Along the way I hope to mention some of: the Mordell-Weil

theorem, the theory of height pairings, special values of L-functions, the Mahler measure of a polynomial, Galois deformations and the motivic

fundamental group.

Mon, 21 Oct 2013

17:00 - 18:00
C5

Finding Galois Representations

Ben Green
Abstract

It is well known that one can attach Galois representations to certain modular forms, it is natural to ask how one might generalise this to produce more Galois representations. One such approach, due to Gross, defines objects called algebraic modular forms on certain types of reductive groups and then conjectures the existence of Galois representations attached to them. In this talk I will outline how for a particular choice of reductive group the conjectured Galois representations exist and are the classical modular Galois representations, thus providing some evidence that this is a good generalisation to consider.

Mon, 14 Oct 2013

17:00 - 18:00
C5

Calculations with elliptic curves

Jan Vonk
Abstract

We will discuss some geometric methods to study Diophantine equations. We focus on the case of elliptic curves and their natural generalisations: Abelian varieties, Calabi-Yau manifolds and hyperelliptic curves. 

Mon, 22 Apr 2013

16:00 - 17:00
SR1

The eigencurve

Jan Vonk
(Oxford)
Mon, 04 Mar 2013

16:00 - 17:00
SR1

A primer on Burgess bounds

Lillian Pierce
(Oxford)
Abstract

We'll present a proof of the basic Burgess bound for short character sums, following the simplified presentation of Gallagher and Montgomery.

Mon, 11 Feb 2013

16:00 - 17:00
SR1

TBC

Netan Dogra
(Oxford)
Mon, 26 Nov 2012

16:00 - 17:00
SR1

Once Upon a Time in Egypt: How the Story of Rational Points Began

Simon Myerson
(Oxford)
Abstract

A nice bed-time story to end the term. It is often said that ideas like the group law or isogenies on elliptic curves were 'known to Fermat' or are 'found
in Diophantus', but this is rarely properly explained. I will discuss the first work on rational points on curves from the point of view of modern number
theory, asking if it really did anticipate the methods we use today.

Mon, 15 Oct 2012

16:00 - 17:00
SR1

Simultaneous prime values of pairs of quadratic forms

Lillian Pierce
(Oxford)
Abstract

Given a form $F(x)$, the circle method is frequently used to provide an asymptotic for the number of representations of a fixed integer $N$ by $F(x)$. However, it can also be used to prove results of a different flavor, such as showing that almost every number (in a certain sense) has at least one representation by $F(x)$. In joint work with Roger Heath-Brown, we have recently considered a 2-dimensional version of such a problem. Given two quadratic forms $Q_1$ and $Q_2$, we ask whether almost every integer (in a certain sense) is simultaneously represented by $Q_1$ and $Q_2$. Under a modest geometric assumption, we are able to prove such a result if the forms are in $5$ variables or more. In particular, we show that any two such quadratic forms must simultaneously attain prime values infinitely often. In this seminar, we will review the circle method, introduce the idea of a Kloosterman refinement, and investigate how such "almost all" results may be proved.


Mon, 28 May 2012

16:00 - 17:00
SR1

The congruent number problem

Frank Gounelas
Abstract

Which positive integers are the area of a right angled triangle with rational sides? In this talk I will discuss this classical problem, its reformulation in terms of rational points on elliptic curves and Tunnell's theorem which gives a complete solution to this problem assuming the Birch and Swinnerton-Dyer conjecture.

Mon, 07 May 2012

16:00 - 17:00
SR1

p-adic zeta functions, p-adic polylogarithms and fundamental groups

Netan Dogra
Abstract

This talk will attempt to say something about the p-adic zeta function, a p-adic analytic object which encodes information about Galois cohomology of Tate twists in its special values. We first explain the construction of the p-adic zeta function, via p-adic Fourier theory. Then, after saying something about Coleman integration, we will explain the interpretation of special values of the p-adic zeta function as limiting values of p-adic polylogarithms, in analogy with the Archimedean case. Finally, we will explore the consequences for the de Rham and etale fundamental groupoids of the projective line minus three points.

Mon, 30 Apr 2012

16:00 - 17:00
SR1

Vinogradov's Three Prime Theorem

James Maynard
Abstract

Vinogradov's three prime theorem resolves the weak Goldbach conjecture for sufficiently large integers. We discuss some of the ideas behind the proof, and discuss some of the obstacles to completing a proof of the odd goldbach conjecture.

Mon, 20 Feb 2012

16:00 - 17:00
SR1

Kloostermania

Alastair Irving
Mon, 13 Feb 2012

16:00 - 17:00
SR1

An introduction to p-adic cohomology

Jan Tuitman
Abstract

In this talk we will give an introduction to the theory of p-adic (or rigid) cohomology. We will first define the theory for smooth affine varieties, then sketch the definition in general, next compute a simple example, and finally discuss some applications.

Mon, 06 Feb 2012

16:00 - 17:00
SR1

Some Galois groups over Q

Jan Vonk
Abstract

The infamous inverse Galois problem asks whether or not every finite group can be realised as a Galois group over Q. We will see some techniques that have been developed to attack it, and will soon end up in the realms of class field theory, étale fundamental groups and modular representations. We will give some concrete examples and outline the so called 'rigidity method'. 

Mon, 30 Jan 2012

16:00 - 17:00
SR1

The Selberg Class - An Introduction

Daniel Kotzen
Abstract

I will discuss the structure of the Selberg class - in which certain expected properties of Dirichlet series and L-functions are axiomatised - along with the numerous interesting conjectures concerning the Dirichlet series in the Selberg class. Furthermore, results regarding the degree of the elements in the Selberg class shall be explored, culminating in the recent work of Kaczorowski and Perelli in which they prove the absence of elements with degree between one and two.

Mon, 23 Jan 2012

16:00 - 17:00
SR1

On the prime k-tuples conjecture

James Maynard
Abstract

We consider the prime k-tuples conjecture, which predicts that a system of linear forms are simultaneously prime infinitely often, provided that there are no obvious obstructions. We discuss some motivations for this and some progress towards proving weakened forms of the conjecture.

Mon, 21 Nov 2011

16:00 - 17:00
SR1

P-adic L-functions and their special values

Netan Dogra
Abstract

This talk will begin by recalling classical facts about the relationship between values of the Riemann zeta function at negative integers and the arithmetic of cyclotomic extensions of the rational numbers. We will then consider a generalisation of this theory due to Iwasawa, and along the way we shall define the p-adic Riemann zeta function. Time permitting, I will also say something about what zeta values at positive integers have to do with the fundamental group of the projective line minus three points

Mon, 24 Oct 2011

16:00 - 17:00
SR1

Radix conversion for polynomials

Sebastian Pancratz
Abstract

We describe various approaches to the problem of expressing a polynomial $f(x) = \sum_{i=0}^{m} a_i x^i$ in terms of a different radix $r(x)$ as $f(x) = \sum_{j=0}^{n} b_j(x) r(x)^j$ with $0 \leq \deg(b_j) < \deg(r)$. Two approaches, the naive repeated division by $r(x)$ and the divide and conquer strategy, are well known. We also describe an approach based on the use of precomputed Newton inverses, which appears to offer significant practical improvements. A potential application of interest to number theorists is the fibration method for point counting, in current implementations of which the runtime is typically dominated by radix conversions.

Mon, 17 Oct 2011

16:00 - 17:00
SR1

On Maeda's conjecture

Jan Vonk
Abstract

The theory of modular forms owes in many ways lots of its results to the existence of the Hecke operators and their nice properties. However, even acting on modular forms of level 1, lots of basic questions remain unresolved. We will describe and prove some known properties of the Hecke operators, and state Maeda's conjecture. This conjecture, if true, has many deep consequences in the theory. In particular, we will indicate how it implies the nonvanishing of certain L-functions.

Mon, 10 Oct 2011

16:00 - 17:00
SR1

Small Gaps Between Primes

James Maynard
(Oxford)
Abstract

We discuss conjectures and results concerning small gaps between primes. In particular, we consider the work of Goldston, Pintz and Yildrim which shows that infinitely often there are gaps which have size an arbitrarily small proportion of the average gap.

Tue, 21 Jun 2011

14:00 - 15:00
L1

An introduction to integer factorization

Jan Tuitman
(Oxford)
Abstract

(Note change in time and location)

The purpose of this talk is to give an introduction to the theory and

practice of integer factorization. More precisely, I plan to talk about the

p-1 method, the elliptic curve method, the quadratic sieve, and if time

permits the number field sieve.

Mon, 09 May 2011

16:00 - 17:00
SR1

163

Frank Gounelas
(Oxford)
Abstract

I will describe why e^{\pi\sqrt{163}} is almost an integer and how this is related to Q(\sqrt{-163}) having class number one and why n^2-n+41 is prime for n=0,...,39. Bits and pieces about Gauss's class number problem, Heegner numbers, the j-invariant and complex multiplication on elliptic curves will be discussed along the way.