Automorphic Forms and Number Theory

我的研究主要集中在自同态形式和l函数，以及它们在数论和朗兰兹程序中的应用。我有四个项目。首先，我们研究了一类数域的数论结果。文献中有许多结果都假设了广义黎曼假设（GRH）。一个人不能无条件地为个别成员证明结果。然而，如果我们考虑一个族，我们可以证明以下两种形式的无条件结果：(1)一个人可以证明一个族的平均结果；(2)除密度零集外，结果对几乎所有成员都成立。第二，单基因数域计数。单基因数域是指具有幂积分基的数域。单基因s_3数域的计数与椭圆曲线的计数有关。即使较弱的结果|L^{(m)}(X)|=o(X)也是未知的。最近Bhargava等人得到了下界|L^{(m)}(X)|>> X^{1/2+1/n}。我得到了二面体扩展的一个结果。设L_d(X)是具有绝对判别式> |d_K|^{1/16}的d_4扩展的集合。这里使用有效的丢番图近似来获得二进制三次形式的最小值。第三，我们研究了朗兰函数，特别是池田升降机。与T. Yamauchi一起，得到了作用于C^{27}内异常管域的q秩为3的E_{7,3}型异常群（E_7型异常群）的一个Ikeda型提升。我们将关于例外群的结果推广到同余子群。此外，我们正在对G_2型特殊组的池田升降机进行研究。这将是第一个四元数离散级数一级尖头形式的例子。我们还可以在E_{6,2}上构造作用于厄米对称域的例外群的一种形式——Ikeda举升。第四，Sp_{2r}全纯尖形的等分布。与S. Wakatsuki和T. Yamauchi一起，获得了GSp_4全纯顶点形式的等分布定理，如垂直的Sato-Tate定理、4次l函数和5次标准l函数的低洼零点。特别是，我们证明了n能级密度与Katz-Sarnak预测一致。我们将结果扩展到Sp_{2r}。特别是，我们可以得到2r+1次标准l函数的n级密度。

My research centers on automorphic forms and L-functions, and their applications in Number Theory and Langlands program. I have four projects.First, we study number theoretic results in a family of number fields. There are many results in the literature assuming generalized Riemann hypothesis (GRH). One cannot prove a result for individual member unconditionally. However, if we consider a family, we can prove unconditional results of the following two forms:(1) one can prove average result in a family; (2) one can prove that the result is valid for almost all members except for a density zero set.Second, counting monogenic number fields. Monogenic number fields are those with power integral basis. Counting monogenic S_3-number fields is related to counting elliptic curves. Even the weaker result |L^{(m)}(X)|=o(X) is unknown. Recently Bhargava and others obtained the lower bound |L^{(m)}(X)|>> X^{1/2+1/n}. I obtained a result for dihedral extensions. Let L_d(X) be the set of D_4-extensions with absolute discriminant> |d_K|^{1/16}. Here effective Diophantine approximation was used to obtain the minimum of binary cubic forms.Third, we study Langlands functoriality, in particular Ikeda lift. With T. Yamauchi, I obtained an Ikeda type lift for exceptional group of type E_{7,3} (exceptional group of type E_7 of Q-rank 3 which acts on the exceptional tube domain inside C^{27}. We will extend our result on the exceptional group to congruence subgroups. Furthermore, we are working on Ikeda lift on the exceptional group of type G_2. It will be the first example of cusp forms with level one with quaternionic discrete series. Also we can construct Ikeda lift on E_{6,2}, a form of the exceptional group which acts on Hermitian symmetric domain.Fourth, equidistribution of holomorphic cusp forms of Sp_{2r}. With S. Wakatsuki and T. Yamauchi, I obtained equidistribution theorems of holomorphic cusp forms of GSp_4 such as vertical Sato-Tate theorem, low-lying zeros of degree 4 L-functions and degree 5 standard L-functions. In particular, we showed that the n-level densities agree with Katz-Sarnak prediction. We are extending our results to Sp_{2r}. In particular, we can obtain n-level density of the degree 2r+1 standard L-functions.