L - functions in Number Theory

L-functions in Number Theory Yichao Zhang Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2010 As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and the connections between objects in different fields. Although most generalizations are largely conjectural, there are many existing results that provide us the evidence. In this thesis, we shall consider some L-functions and look into some problems mentioned above. More explicitly, for the L-functions associated to newforms of fixed squarefree level, we will consider an average version of the fourth moments problem. The final bound is proven by considering definite rational quaternion algebras and divisor functions in them, generalizing Maass Converse Theorem and one of Duke’s result and eventually applying the solution to Basis Problem. We then consider the problem of expressing the special value at 1 2 of the RankinSelberg L-function associated to two newforms in terms of the Pertersson inner product, where one of the newforms is twisted by the derivative of some Eisenstein series. Finally, we consider the Artin L-functions attached to irreducible 4-dimensional S5Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong Artin Conjecture in this case.