Number Theory

Research in number theory within CeCANT includes both analytic and algebraic number theory, with emphases on modular forms, Maass forms, Siegel modular forms, representations of p-adic groups, the Langlands programme, L-functions, Galois representations, and arithmetic quantum chaos.

A key area of our research strength is bridging representation theory and analytic number theory, an area of tremendous current worldwide interest with a Fields medal awarded to Akshay Venkatesh in 2018. Big recent successes have been made in understanding analytic properties of higher rank L-functions, including by Prof Saha and Dr Jana, the former funded by the Leverhulme Trust and EPSRC.  In 2023, Prof Saha hosted the largest ever conference in the UK on analytic aspects of automorphic forms and L-functions of higher rank.

Another area of number theory our group represents is the Langlands programme, a central subject in algebraic number theory connecting it with algebraic geometry, representation theory and analysis. One of our members, Dr Sasaki, is an academic descendant of A. Wiles (famous for the proof of Fermat's Last Theorem) with expertise in arithmetic geometry, and has made major contributions to Artin's conjecture and the Serre weight conjecture (with F. Diamond).

Covering of SL₂(ℤ)\ℍ² by balls around SL₂(ℤ)\SL₂(ℤ[1/3])