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

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(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).