The aim of the project is to explore the usefulness of Taylor-Wiles systems for establishing cases of the Tamagawa number conjecture on special values of Hasse-Weil L-functions. Following the initial breakthrough by Taylor and Wiles this has been done in the case of the adjoint L-function of elliptic modular forms in joint work with Diamond and Guo. The next cases we intend to look at are those where the Taylor-Wiles method has been worked out but the relationship to values of L-functions is still missing, notably Siegel modular forms of genus 2, Hilbert modular forms and possibly unitary groups.
L-functions figure prominently in modern number theory, or even in all of pure mathematics if one takes as a benchmark the seven Clay millenium problems, two of which are directly concerned with L-functions. In high school algebra one draws the solution set in the plane of an algebraic equation in two variables x and y. One may also look at the solutions in rational numbers, integers or integers modulo a prime number. L-functions are built from the number of solutions modulo primes (of any set of equations in any number of variables) and are expected to give information about solutions in rational numbers. Solving equations in rational numbers is notoriously hard, whereas values of L-functions are often very computable, so such a relationship lies very deep. The primordial example is of course the conjecture of Birch and Swinnerton-Dyer, one of the millenium problems. As often in mathematics, one generalizes a problem in order to understand it better but the tradeoff is an increasingly abstract framework ("cohomology" in this case). On the positive side, other instances of the generalized framework may actually be provable with current methods, and this is what the project aims to explore. The aim is to prove new cases of the expected special value formula for L-functions, using the relatively recent method of Taylor-Wiles systems.
