UC Berkeley

This thesis is a combination of three pieces of work: 1, We explore some axioms of divergent series and their relations with conformal field theory. As a consequence we obtain another way of calculating $L(0,\chi)$ and $L(-1,\chi)$ for $\chi$ being a Dirichlet character. We hope this discussion is also of interest to physicists doing renormalization theory for a reason indicated in the Introduction section. We consider a twist of the oscillator representation of the Virasoro algebra by a group of Dirichlet characters first introduced by Bloch in \cite{Bloch}, and use this to give a 'physical interpretation' of why the values of certain divergent series should be given by special L values. Furthermore, we use this to show that some fractional powers which are crucial for some infinite products to have peculiar modular transformation properties are expressed explicitly by certain linear combinations of $L(-1, \chi)$'s for appropriately chosen $\chi$'s, and can be understood physically as a kind of 'vacuum Casimir energy' in our settings. 2, We give an attempt to reinterpret Tate's thesis by a sort of conformal field theory on a number field. Based on this and the existence of a hypothetical 3-dimensional gauge theory, we give a physical interpretation of the Gauss quadratic reciprocity law by a sort of S-duality. 3, We find a complete list of positive definite symmetric matrices with integer entries $\begin{bmatrix}a&b\b&d\end{bmatrix}$ such that all complex solutions to the system of equations

\begin{align}

1-x_1=x_1^ax_2^b\nonumber \

1-x_2=x_1^bx_2^d\nonumber

\end{align}

are real. This result is related to Nahm's conjecture in rank 2 case. (In each chapter, except for equations, chapter number is omitted from quotation. For example, theorem 1.1.1 will be quoted as theorem 1.1.)