Investigation of zeta functions associated with prehomogeneous vector spaces

Let L be the lattice of integral binary cubic forms and L I D4^I D 4 be the dual lattice of L.The distribution of cubic fields is closely related to the prehomogeneous vector space of binary cubic forms.The zeta functionsξD 2 i i D 2(L，s)(I=1，2)associated with this space are expressed as sums of|D I D 2 K I D 2|I D 1 s D 1ηD 2 K D 2(2s)over all cubic fields K.HereηD 2 K I D 2(S)=ζ(2s)ζ(3 s-1)I D 7ζD 2 K E D2(S)(/)ζD2(2 S)D 2.Using this expression and class field theory，I proved Ohno conjecture which statesξD21个D2(L I D4^I D4，s)=3个D1-3s个D1ξD22个D2(L，s)andξD22个D2(L，s)=3个D11-3s个D1ξD21个D2(L，s)。As applications of this result，I obtained certain relations among the number of cubic fields of positive and negative discriminants，and a refinement of Sholz‘s reflection theorem.These results are published in Inventiones mathematicae.I also gave a talk on the results at International Congress of Mathematicians ICM98.I have been studying the prehomogeneous vector spaces of pairs of ternary quadratic forms which is closely related to the distribution of discriminants of quartic fields and 2-torsion subgroups of ideal class groups of cubic fields.In particular，I have obtained certain relations between the set of equivalence classes of pairs of integral ternary quadratic forms and 2-torsion subgroups of cubic fields。I gave a talk on this result at the symposium on number theory at Tsuda College.

Let L be the lattice of integral binary cubic forms and LィイD4^ィエD4 be the dual lattice of L. The distribution of cubic fields is closely related to the prehomogeneous vector space of binary cubic forms. The zeta functions ξィイD2iィエD2(L, s)(I = 1, 2) associated with this space are expressed as sums of |DィイD2KィエD2|ィイD1sィエD1ηィイD2KィエD2(2s) over all cubic fields K. Here ηィイD2KィエD2(s) =ζ(2s)ζ(3s - 1)ィイD7ζィイD2KィエD2(s)(/)ζィイD2KィエD2(2s)ィエD7. Using this expression and class field theory, I proved Ohno conjecture which statesξィイD21ィエD2(LィイD4^ィエD4, s) = 3ィイD1-3sィエD1ξィイD22ィエD2(L, s) andξィイD22ィエD2(LィイD4^ィエD4, s) = 3ィイD11-3sィエD1ξィイD21ィエD2(L, s). As applications of this result, I obtained certain relations among the number of cubic fields of positive and negative discriminants, and a refinement of Sholz's reflection theorem. These results are published in Inventiones mathematicae. I also gave a talk on the results at International Congress of Mathematicians ICM98.I have been studying the prehomogeneous vector spaces of pairs of ternary quadratic forms which is closely related to the distribution of discriminants of quartic fields and 2-torsion subgroups of ideal class groups of cubic fields. In particular, I have obtained certain relations between the set of equivalence classes of pairs of integral ternary quadratic forms and 2-torsion subgroups of cubic fields. I gave a talk on this result at the symposium on number theory at Tsuda College.