On exceptional automorphic forms in classical problems of analytic number theory

This project falls within the EPSRC Number theory research area.
Analytic number theory studies quantitative properties of the integers, prime numbers, and related arithmetic objects, using a variety of methods ranging from complex, harmonic, and functional analysis to probability and discrete mathematics. Besides the rich diversity of mathematical techniques involved, analytic number theory stands out through its great number of unsolved problems, many of which have easy-to-grasp statements. For instance, it remains completely open whether there exist infinitely many pairs of twin primes (differing by exactly two), or infinitely many primes that equal a perfect square plus one. Even counting the primes up to a given threshold, with a small error term, turns out to be extremely difficult; this is the topic of the infamous Riemann hypothesis. The methods developed to attack such problems often find applications to other branches of mathematics, and related fields like cryptography.
The main goal of this project is to study classical arithmetic problems, using modern techniques from the analytic theory of automorphic forms. Roughly speaking, automorphic forms are central objects of interest in number theory, which obey certain symmetries and share additive and multiplicative structure; they generalize classical Dirichlet characters as well as modular forms. In analytic problems, automorphic forms often arise as a tool to prove cancellation in exponential sums, which formalizes certain pseudo-randomness properties. Unfortunately, many properties of automorphic forms remain conjectural, and a recurring theme in this project is showing that there are few `exceptional' forms that fail these conjectures, which leads to unconditional results. As tools, we use a combination of spectral methods (based on trace formulae) and automorphic L-functions (which generalize the classical Riemann zeta function, and have links to Galois representations via the Langlands program).