# Sheffield Number Theory Group

## About the Sheffield Number Theory group

The number theory group has six permanent academic staff, and currently one postdoc and six PhD students.

Basic tools are number fields (finite extensions of the rationals, generated by roots of polynomials with rational coefficients) and their local completions such as p-adic fields (which package congruences modulo powers of a prime p).Â

A central focus of our interests is the representation of Galois groups by homomorphisms to groups of matrices. Such Galois representations arise from the action of Galois groups on topological invariants of spaces defined by polynomial equations in several variables with rational coefficients (for example ``elliptic'' curves y^2=x^3+ax+b), so algebraic geometry is important here.

Other key players are automorphic forms and associated automorphic representations, local components of which are representations of p-adic, or real, matrix groups as linear transformations on infinite dimensional spaces of functions. Certain analytic functions called L-functions, generalising the Riemann zeta function, are also important. Still, what we do is perhaps better classified as algebraic, rather than analytic, number theory.

Influential conjectures of Langlands link Galois representations with automorphic forms (and different types of automorphic forms with each other), saying that their associated L-functions are the same. An instance is the modularity of elliptic curves, implicated in the proof of Fermat's Last Theorem. But the most basic example is the quadratic reciprocity law, when viewed the right way!

Recent PhD projects include ones on representations of p-adic groups, applications of modularity to diophantine equations, deformation theory of Galois representations, computation of automorphic forms, and congruences between automorphic forms and their connections with values of L-functions at integer points. Availability of supervisors will vary from year to year.