Quadratic Forms over Global Fields

Abstract

This thesis is structured in two parts. The first part explores certain binary quadratic forms over the polynomial ring $\mathbb{F}_q[T]$. We derive explicit formulas for the number of representations of a polynomial and estimate their moments in two asymptotic scenarios: the large finite field limit, where the field size $q$ grows with fixed polynomial degree $n$, and the large degree limit, where the degree $n$ increases while $q$ remains fixed. In the former, we employ a Dirichlet series framework to extract asymptotic behavior, while in the latter, we apply a refined partitioning of the space of polynomials of fixed degree to obtain sharp asymptotic estimates. The second part investigates the representation of integers as sums of an even number of triangular numbers. Using the Hardy–Littlewood circle method, we sum the associated singular series and establish its convergence to the Eisenstein component of the expressions obtained using the theory of modular forms, which are expressed in terms of generalized divisor functions.