**Number Theory Session**

*Talks will be in MR5.*

**Thursday**

- 2.30-3.15: Graham Sills - Height bounds on n-coverings
- 3.15-4.00: Mohammad Sadek - Models of binary quartic curves
*Tea/Coffee*- 4.30-5.15: Rachel Newton - The local reciprocity map
- 5.15-6.00: Graeme Taylor - Cyclotomic matrices and graphs

**Friday**

*Plenary lecture, Tea/Coffee*- 11.00-11.45: Linda Gruendken - Semi-direct product Galois covers of curves in characteristic $p$ (PDF)
- 11.45-12.30: James Newton - Level raising for p-adic modular forms
*Lunch, Panel discussion, Tea/Coffee*- 4.00-4.45: Joel Cohen - An introduction to random matrices and zeta functions (accessible talk)
- 4.45-5.30: Ivan Fesenko (keynote speaker) - Modern number theory as unifying factor for mathematics

*Height bounds on n-coverings* - Graham Sills

I shall discuss some background theory of Elliptic Curves including the concepts of Two and Four-Descent. Then I will introduce heights and we will see how it is useful to consider local heights on Binary Quartics and Quaternary Quadratic Forms in order to search for the generators of an Elliptic Curve. There may also be some discussion of the algorithm for implementation.

*Models of binary quartic curves* - Mohammad Sadek

By a binary quartic curve we mean a curve of the form C: y^2 = a x^4 + b x^3 + c x^2 + d x + e. We define what we mean by a minimal model of C, our definition can be considered as a generalisation of the terminology for a minimal Weierstrass model of an elliptic curve, but minimal models of binary quartic curves are far from being unique up to isomorphism. We introduce geometric criteria for any model of C to be minimal and then explain how to count minimal models up to isomorphism.

*The local reciprocity map* - Rachel Newton

I will take the old-fashioned approach and develop a definition of the local reciprocity map (for finite abelian extensions of the p-adics) using central simple algebras and cyclic algebras. Along the way, I will introduce the Brauer group (which classifies central simple algebras) and define the Hasse invariant. I will describe the cases in which an explicit formula has been found for the local reciprocity map and try to explain why this is difficult in general for totally ramified extensions. This talk should be accessible to anyone who has encountered extensions of local fields.

*Cyclotomic matrices and graphs* - Graeme Taylor

Lehmer's Conjecture, an open problem in number theory, motivates the study of 'cyclotomic' matrices - those with all eigenvalues in the interval [-2,2]. Under the additional restriction of integer entries, such matrices can be associated with charged signed graphs, ultimately leading to their classification. I will discuss this result, as well as my own generalisation to cyclotomic matrices/graphs over rings of integers of imaginary quadratic extensions.

*Semi-direct product Galois covers of curves in characteristic $p$* - Linda Gruendken

Let $k$ be an algebraically closed field with characteristic $p>0$. A result of Abhyankar shows that the algebraic fundamental group of the affine line $\bba^1_k$ over $k$ is contained in the set of quasi $p$-groups. In joint work with R. Priez, K. STevenson, E. Ozman et al., I consider $\Gb$-covers of $\bbp^1_k$ ramified only at $\infty$ and find the minimal genus of such a cover and the number of these covers. We show that the minimal genus depends only on $\ell$ and $p$ and the number of these covers equals the order of $\ell$ modulo $p$, and this talk aims to give an overview of these results.

*Level raising for p-adic modular forms* - James Newton

In the early 1970s Serre constructed a p-adic analytic family of modular forms, interpolating the classical Eisenstein series. This idea was gradually expanded, until in the 1990s Coleman and Mazur showed that all modular forms of "finite slope" lie in p-adic families, and constructed a geometric object (the "eigencurve") parametrising these p-adic modular forms. In this talk I will give an overview of some of the theory of p-adic modular forms and discuss a level raising result on the eigencurve.

*An introduction to random matrices and zeta functions* (accessible talk) - Joel Cohen

The aim of this talk is to give a brief introduction to the links between the distribution of spacings between zeros of the Riemann zeta function (and L-functions) and random matrices. We first give an overview of random matrices, and especially the computation of asymptotical distribution of eigenvalues. We then study theoretical and numerical evidence that spacings between zeros follow similar laws.

*Modern number theory as unifying factor for mathematics* - Ivan Fesenko (keynote speaker)

Number theorists use in their research almost all areas of pure mathematics,
and several fundamental developments in number theory serve as a unifying
force in mathematics.

I will present several recent instances of fruitful interplay between number
theory and algebra, geometry, topology, functional analysis and emerging new
links to quantum physics.