Browsing by Author "Getz, Jayce R"

Conjectures of Braverman-Kazhdan, L. Lafforgue, Ngô and Sakellaridis imply that all affine spherical varieties admit generalized Poisson summation formula. In this dissertation we establish a generalized Poisson summation formula for certain spaces of test functions on the zero locus of a quadratic form. The functions are built from the Whittaker coefficients of automorphic representations on GL_n. We also give an expression of the local factors where all the data is unramified.

This thesis aims to study the relative theory of endoscopy. Let $F$ be a nonarchimedean local field and consider the reductive group $\mathrm{O}_3(F)$ acting on the symmetric variety $\mathcal{S}(F)=(\mathrm{U}_3/\mathrm{O}_3)(F)$ over $F$. We compute the endoscopic orbital integrals of the basic function in this situation. Knowledge on the endoscopic orbital integrals is essential for observing the existence of endoscopy groups and transfer of smooth functions in the relative setting. This would be the first time such a computation has appeared in the literature for a spherical variety with type $N$-spherical roots.

Let $F$ be a non-degenerate quadratic form on an $n$-dimensional vector space $V$ over the rational numbers, and let $J$ be the symmetric matrix associated to $F$. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size $B$. Heath-Brown gave an asymptotic for this question of the form: $c_1 B^{n-2} + O_{J,\epsilon}(B^{(n-1+\delta)/2+\epsilon})$, for any $\epsilon > 0$ and dim$V \geq 5$, where $c_1 \in \mathbb{C}$ and $\delta=0$ or $1$, according as $n$ is odd or even. For dim$V = 3$ and dim$V = 4$, Heath-Brown also gave similar asymptotics. More recently, Getz gave an asymptotic of the form: $c_1 B^{n-2} + c_2 B^{n/2} + O_{J,\epsilon}(B^{n/2+\epsilon-1})$ when $n$ is even, in which $c_2 \in \mathbb{C}$ has a pleasant geometric interpretation. We consider the case where $n$ is odd with diagonal unimodular $J$ and give an analogous asymptotic of the form: $c_1 B^{n-2} + c_2 B^{(n-1)/2} + O_{J,\epsilon}(B^{n/2+\epsilon-1})$. We use the circle method and work classically to exploit Gauss sums and find Dirichlet characters that fit into the odd degree case. We also provide an explicit description of the Dirichlet series arisen during the investigation, which is useful in applications. It turns out that the geometric interpretation of the constant $c_2$ of the asymptotic in the odd degree and even degree cases is strikingly different.