Departmental PhD Thesis Exam – Yichao Zhang

Wednesday, May 5, 2010, 12:30-1:30 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Yichao Zhang

PhD Advisor:    Henry Kim

Thesis title:  L-functions in number theory
http://www.math.toronto.edu/dept/Zhang_Yichao_thesis.pdf

Thesis Abstract:

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 square-free 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
Rankin-Selberg 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
$S_5$-Galois 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.