Automorphic Forms and Number Theory

我主要研究自同构型和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，我们研究了一族数域中的数论结果。在假设广义黎曼假设(GRH)的文献中有很多结果。一个人不能无条件地证明个别成员的结果。然而，如果我们考虑一个家庭，我们可以证明以下两种形式的无条件结果： (1)可以证明一个族中的平均结果；(2)可以证明该结果对除密度零集以外的几乎所有成员都有效。 二是对单基因数域进行计数。一元数域是指具有幂整数基的数域。对单基因S 3数域的计数与对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)|&gt；&gt；X^{1/2 1/n}。我得到了二面体扩张的一个结果。设L_d(X)是具有绝对判别式的D_4-扩张集，|d_K|^{1/16}。这里使用有效丢番图近似来获得二元三次形的最小值。 第三，我们研究了朗兰兹函数，特别是Ikeda Lift。与T.Yamuchi一起，我得到了E_(7，3)型例外群(作用于C^{27}内例外管域上的Q-秩3型E_7型例外群)的Ikeda型升力。我们将把我们关于例外群的结果推广到同余子群。此外，我们还研究了G_2型例外群上的Ikeda升力。这将是第一个具有四元数离散级数的一级尖点形式的例子。我们还可以在E_(6，2)上构造Ikeda Lift，E_(6，2)是作用在Hermite对称域上的例外群的一种形式。 第四，Sp_(2r)的全纯尖点形式的均匀分布。与S.Wakatsuki和T.Yamuchi一起，我得到了GSP_4的全纯尖点形式的等分布定理，如垂直Sato-Tate定理、四次L函数的低阶零点和五次标准L函数。特别地，我们证明了n能级密度与Katz-Sarnak的预测一致。我们将我们的结果推广到Sp_{2R}。特别地，我们可以得到2R1次标准L-函数的n能级密度。

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.