L-functions and Arithmetic
Presented by Professor James Brown, Clemson University
One of the main themes in number theory can be summarized as follows: Given a number theoretic object X of interest, one can define an associated complex analytic function L(s,X) by using only ''local'' data about X. One can then study this L-function using complex analysis to arrive at amazing theorems and conjectures about X. I will explain such a conjectural relationship about elliptic curves in the undergraduate talk. I will begin this talk by stating a theorem along these lines in the case X is a number field and then recalling the statement about elliptic curves. I will then focus on a generalization of this conjecture to the setting of modular forms and what can be said in this setting.