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.)

We describe an algorithm to compute bases of modular forms with rational coefficients for the Weil representation associated to an even lattice. In large enough weights the forms we construct are zero-values of Jacobi forms of rational index, while in smaller weights our construction uses the theory of mock modular forms. The main application is in computing automorphic products.

We will discuss several questions related to q-hypergeometric series and modular functions. Nahm's conjecture is an attempt to answer the question of when a certain q-hypergeometric series can have modularity. With this conjecture as motivation, we will consider a few closely related algebraic objects such as Bloch groups, Y-systems, T-systems, Q-systems, the dilogarithm function and its variants.

In this article, we give a tangle approach in the study of Legendrian knots in the standard

contact three-space. On the one hand, we define and construct Legenrian isotopy invariants

including ruling polynomials and Legendrian contact homology differential graded algebras

(LCH DGAs) for Legendrian tangles, generalizing those of Legendrian knots. Ruling polynomials

are the Legendrian analogues of Jones polynomials in topological knot theory, in the

sense that they satisfy the composition axiom.

On the other hand, we study certain aspects of the Hodge theory of the “representation

varieties (of rank 1)” of the LCH DGAs, called augmentation varieties, associated to Legendrian

tangles. The augmentation variety (with fixed boundary conditions), hence its mixed

Hodge structure on the compactly supported cohomology, is a Legendrian isotopy invariant

up to a normalization. This gives a generalization of ruling polynomials in the following

sense: the point-counting/weight (or E-) polynomial of the variety, up to a normalized factor,

is the ruling polynomial. This tangle approach in particular provides a generalization

and a more natural proof to the previous known results of M.Henry and D.Rutherford. It

also leads naturally to a ruling decomposition of this variety, which then induces a spectral

sequence converging to the MHS. As some applications, we show that the variety is of Hodge-Tate

type, show a vanishing result on its cohomology, and provide an example-computation

of the MHSs.