Automorphic Forms and Number Theory

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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.