Arithmetic of definite shifted lattices and universal sums of generalized polygonal numbers

Abstract

The thesis consists of two parts. In the first part, we study the arithmetic of positive-definite shifted lattices. Let $X$ be a positive-definite integral shifted lattice of rank $r$ over $\mathbb{Q}$. We are interested in obtaining an exact or asymptotic formula for the number of representations of integers by $X$. The main tool is the interplay between the arithmetic theory of shifted lattices and the theory of modular forms. Because the theta series associated to a positive-definite integral shifted lattice is a modular form, it splits into a sum of an Eisenstein series and a cusp form. When $r=3$, the cusp form further splits into a sum of unary theta functions and a cusp form orthogonal to unary theta functions. By generalizing Eichler commutation relation to the cases of shifted lattices, the splitting can be interpreted in terms of the arithmetic information of the shifted lattice. In particular, the Eisenstein part is given by the theta series associated to the proper genus of $X$. By Siegel--Minkowski formula, the Fourier coefficient of the Eisenstein part is a product of local densities. We generalize Yang's explicit formulae for local densities to the cases of shifted lattices to evaluate the Eisenstein part. Then we can give asymptotic formulae for the number of representations. To obtain an exact formula, we study the growth of proper class numbers of shifted lattices because if the proper class number is one, the number of representations is equal to the Fourier coefficient of the Eisenstein part, which can be calculated by evaluating the local densities using explicit formulae. To study the proper class number, we establish an exact formula and a lower bound for the proper mass. Then it follows that there are finitely many positive-definite shifted lattices of rank $r=2$ with a primitive lattice part and a fixed proper class number.

In the second part, we apply the arithmetic theory of shifted lattices to study universal sums of generalized polygonal numbers. To classify universal sums, we prove finiteness theorems showing that any sum $\sum_{i=1}^{r}a_iP_{m_i}$ such that $\text{lcm}(m_1-2,\ldots,m_r-2)\leq\mathfrak{M}$ is universal if it represents positive integers up to a constant $\Gamma_{\mathfrak{M}}$ for any integer $\mathfrak{M}\geq1$. An asymptotic upper bound for the constant $\Gamma_{\mathfrak{M}}$ is given with an ineffective implied constant. Though the implied constant of the upper bound is ineffective, the constant $\Gamma_{\mathfrak{M}}$ is effective for small values of $\mathfrak{M}$. In a paper of Bosma and Kane, they showed that $\Gamma_1=8$. In this thesis, we calculate the next value and show that $\Gamma_2=48$. The implied constant is ineffective because there exist potentially universal ternary sums, but it is difficult to confirm the universality of them. The list of these ternary sums is given in the appendix and we apply Ono and Soundararajan's conditional approach to prove the universality of around $20\%$ of the ternary sums conditional on various generalized Riemann hypotheses.