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

(1)对二元三次型空间的积分模型进行了分类，并证明了相应的zeta函数都满足对偶恒等式（与Y。Ohno和S. Wakatsuki）。(2)我确定了计算三次域的第二主函数，从而解决了罗伯茨猜想。此外，通过引入轨道L函数的概念，我证明了在算术级数中计数立方域存在偏差（与F。Thorne）。(3)在一个正在进行的工作中，我正在研究一个特殊的可约预齐次向量空间的10维。我给出了整数轨道的代数解释，并且还发现，对于具有水平结构的二元三次型空间，二元中相关的zeta函数可以用zeta函数族表示（与G. Chinta）。

(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).