My recent research centers on automorphic forms and L-functions, and their applications in Number Theory and Langlands program.
1. Analytic Number Theory
I 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.
With P.J. Cho, I obtained results of this type for the following: smallest primes in a conjugacy class, logarithmic derivatives of L-functions at s=1.
Extreme values of Dedekind zeta functions at s=1.
Average of the smallest prime.
Logarithmic derivatives of Artin L-functions at s=1.
Universality of Artin L-functions.
2. Langlands functoriality, in particular Ikeda lift.
Langlands functoriality predicts that given an L-group homomorphism from the L-group of H to that of G, there exists a natural map from automorphic representations of H to those of G. A particular case is Ikeda lift, where H=PGL_2 and G=Sp_{4n}. 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}).
Miyawaki type lift for GSpin(2,10).
3. 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. A main tool is Arthur's invariant trace formula. While Shin-Templier used Euler-Poincar´e functions at infinity in the formula, we use a pseudocoefficient of a holomorphic discrete series to isolate holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms in the spectral side.
Equidistribution of holomorphic Siegel cusp forms of degree 2.
Equidistribution of holomorphic Siegel cusp forms: Hecke fields and n-level density.