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