Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

Theta series encode the number of representations of integers by quadratic forms in their Fourier coefficients and connect classical counting problems to the theory of modular forms and elliptic curves. On the other hand, theta series connect with hypergeometric series via Ramanujan’s theory of elliptic functions to alternative bases. This dissertation explores these connections and applies theta series to solve problems involving representations of integers  and  special values of $L$-functions. We analyze the theta series associated to the quadratic form $Q(\Vec{x})=x_1^2+x_2^2+x_3^2+x_4^2$ with congruence conditions on $x_i$ modulo $2,3,4$ and $6$. By interpreting theta series as modular forms, we can express the number of representations of integer solutions represented by given quadratic forms $Q(\Vec{x})$ in terms of the number of rational points of certain elliptic curves over finite fields. Using the modularity of hypergeometric Galois representations associated with the given hypergeometric series, we also relate special values of $L$-functions of certain Hecke eigenforms with complex multiplication to classical hypergeometric series$ {}_3F_2\!\left[\begin{matrix} \frac{1}{2}\ \frac{1}{d}\ \frac{d-1}{d} \\ \ \ \ 1\ \ \ 1 \end{matrix} \; ; t_d \right]$, for some specific modular function $t_d$ where $d=2,3,4$ and $6$. Moreover, we provide a complete list of all such eigenforms arising from tensor products of representations attached to elliptic curves over real quadratic fields with complex multiplication, together with their associated special $L$-values.

Date

3-27-2026

Committee Chair

Tu, Fang-Ting

LSU Acknowledgement

1

LSU Accessibility Acknowledgment

1

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