Number theory for parabolic type prehomogeneous vector spaces and their associated zeta functions

(1)对二元三次形空间的积分模型进行了分类，证明了相关的Zeta函数都满足对偶恒等式(与Y.Ohno和S.Wakatsuki共同工作).(2)确定了计算三次域的第二主数，从而解决了Roberts猜想.此外，通过引入轨道L函数的概念，我证明了在算术级数中(与F.Thorne共同工作)对立方场的计数是有偏差的。(3)在一项正在进行的工作中，我正在研究一种特殊的可约的十维预齐次向量空间。我给出了整数轨道的代数解释，并且还发现相关联的两个变量的Zeta函数表示为具有水平结构的二元三次形式空间的Zeta函数族(与G.Chinta共同工作)。

(1) I classified the integral models for the space of binary cubic forms, and proved that the associated zeta functions all satisfy the dual identities(joint work with Y. Ohno and S. Wakatsuki).(2) I determined the second main for counting cubic fields and therefore solved the Roberts' conjecture. Moreover, by introducing the notion of orbital L functions, I showed that there are biases for counting cubic fields in arithmetic progressions(joint work with F. Thorne).(3) In an ongoing work, I am studying one particular reducible prehomogeneous vector space of ten dimensional. I gave an algebraic interpretation of integer orbits, and also find that the associated zeta function in two variables are expressed in terms of a family of zeta functions for the space of binary cubic forms with level structures(joint work with G. Chinta).