L2

Iwasawa algebras are completed group rings that arise in number theory, so there is interest in understanding their prime ideals. For some special Iwasawa algebras, it is conjectured that every non-zero such ideal has finite codimension and in order to show this it is enough to establish the faithfulness of the modules arising from the completion of highest weight modules. In this talk we will look at methods for doing this and apply them to the specific case of the metaplectic representation for the symplectic group.

In this seminar, I will talk about a notion of entropy of arithmetic functions and some properties of this entropy.  This notion was introduced to study Sarnak's Moebius Disjointness Conjecture.

Let $E$ be an elliptic curve over a number field $K$ and $n \geq 2$ an integer. We recall that elements of the $n$-Selmer group of $E/K$ can be explicitly written in terms of certain equations for $n$-coverings of $E/K$. Writing the elements in this way is called conducting an explicit $n$-descent. One of the applications of explicit $n$-descent is in finding generators of large height for $E(K)$ and from this point of view one would like to be able to take $n$ as large as possible. General algorithms for explicit $n$-descent exist but become computationally challenging already for $n \geq 5$. In this talk we discuss combining $n$- and $(n+1)$-descents to $n(n+1)$-descent and the role that invariant theory plays in this procedure.

TBC

The Liouville function $\lambda(n)$ is defined to be $+1$ if $n$ is a product of an even number of primes, and $-1$ otherwise. The statistical behaviour of $\lambda$ is intimately connected to the distribution of prime numbers. In many aspects, the Liouville function is expected to behave like a random sequence of $+1$'s and $-1$'s. For example, the two-point Chowla conjecture predicts that the average of $\lambda(n)\lambda(n+1)$ over $n < x$ tends to zero as $x$ goes to infinity. In this talk, I will discuss quantitative bounds for a logarithmic version of this problem.

Many congruences between modular forms (or at least their q-expansions) can be explained by the theory of $p$-adic families of modular forms. In this talk, I will discuss properties of eigenvarieties, a geometric interpretation of the idea of $p$-adic families. In particular, focusing initially on the well-understood case of (elliptic) modular forms, before delving into the considerably murkier world of Bianchi modular forms. In this second case, this work gives numerical verification of a couple of conjectures, including BSD by work of Loeffler and Zerbes.

Modular curves play a key role in the Langlands programme, being the simplest example of so-called Shimura varieties.  Their less famous cousins, Shimura curves, are also very interesting, and very concrete. 
In this talk I will give a gentle introduction to the arithmetic of Shimura curves, with lots of explicit examples. Time permitting, I will say something about recent work about intersection numbers of geodesics on Shimura curves.