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Number Theory

Research

The work carried out at King's spans a variety of topics at the forefront of current number theory research, including L-functions, automorphic forms, Galois representations and the geometry of numbers.

L-functions are certain complex-valued functions that can be associated to arithmetic objects, such as algebraic varieties over number fields. The best known example of an L-function is the Riemann zeta function. Our interest in L-functions centres on their extraordinary capacity for reflecting the fine algebraic structure of the objects to which they are associated. A basic example of this is the analytic class number formula, which relates arithmetic invariants of number fields to values of L-functions. An important direction of current research concerns far-reaching conjectures that vastly generalise this formula, linking fundamental questions in number theory to properties of L-functions and their p-adic valued analogues.

Another important topic of research in the group is the theory of automorphic forms, another class of analytic functions with deep connections to number theory. The best known examples are classical modular forms, whose connection with elliptic curves played a crucial role in the proof of Fermat's Last Theorem. This connection is but a special case in the Langlands Programme, a vast web of conjectures relating automorphic forms to arithmetic objects, such as algebraic varieties and representations of Galois groups. The group's research focuses on establishing and understanding this relationship in various contexts --- local and global, classical and p-adic. This bridge between analytic and arithmetic objects has far-reaching applications in number theory, strongly linked with the role of L-functions described above.

The group also investigates the relation with mathematical physics and its applications. One such relation comes from the correspondence between the integer lattice in R2 and the energy levels of the torus; hence the angular distribution of lattice points determines the directions of wave propagation on the torus. This establishes a connection between fundamental problems in the geometry of numbers and the emerging field of quantum chaos, a meeting point of mathematical physics, number theory and other disciplines. These problems and related ones in physics are pursued with mathematical rigour.

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