论文标题
$ g_2 $上的模块化表格的特殊theta函数和算术性
Exceptional theta functions and arithmeticity of modular forms on $G_2$
论文作者
论文摘要
分裂特殊组上的Quaternionic模块化表格$ G_2 = G_2^S $由Gan-Gross-Savin定义。这些自形功能的一个显着特性是它们具有傅立叶膨胀和傅立叶系数的强大概念，类似于Shimura品种上经典的霍明型模块化形式。在本文中，我们证明，在重量$ \ ell $中，至少$ 6 $，有一个cuspidal模块化形式的重量$ \ ell $的基础，使得此基础元素的所有傅立叶系数都在$ \ \ mthbf {q} $的环形范围内。我们证明这一点的主要工具是在$ g_2 $上开发一个“特殊theta函数”的概念。
Quaternionic modular forms on the split exceptional group $G_2 = G_2^s$ were defined by Gan-Gross-Savin. A remarkable property of these automorphic functions is that they have a robust notion of Fourier expansion and Fourier coefficients, similar to the classical holomorphic modular forms on Shimura varieties. In this paper we prove that in even weight $\ell$ at least $6$, there is a basis of the space of cuspidal modular forms of weight $\ell$ such that all the Fourier coefficients of elements of this basis are in the cyclotomic extension of $\mathbf{Q}$. Our main tool for proving this is to develop a notion of "exceptional theta functions" on $G_2$.
