| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1999: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Research Abstract |
Let K be a totally real algebraic number field. The theta series associated with a positive quadratic form with coefficients in K is a Hilbert modular form. We consider the case that the number of the variables of the quadratic form is small. Then to investigate the numbers of representations by the quadratic form, or the relation between the numbers of the representations by the quadratic form and the special values of Dedekind L-function, we need to investigate the space of Hilbert modular forms of low weight.
The case that K be real quadratic, is mainly treated. It is accomplished to describe reducible loci on the Hilbert modular surface. Let O_K be the maximal order of K, and let O_K be the different. Let λ, η be the Z-base of O_K with Nm(λη)<0, Nm denoting the norm map. Then there are only a finite number of such λ, η up to multiplications by units. Call U, the complete representatives of the classes of (λ, η) modulo the unit group. Let<<numerical formula>>.Then there is the natural one to one correspondence between U and the set of reducible loci on the Hilbert modular surface associated with Γ_<OK>. V.A.Gritsenko and V.V.Nikulin showed that in the Siegel modular case of degree 2, the theta series is written as a infinite product. From moduli theory, there is the natural map of the Hilbert modular surfaces associated with Γ_<OK> into Siegel modular variety of degree two, which is called a modular embedding. Theta series of Hilbert modular case are obtained from theta series of Siegel case via the ring homomorphism associated with the modular embedding. Since theta series on the Hilbert modular surface associated with Γ_<OK> vanishes only at reducible loci, we obtain the infinite product expression of theta series on the Hilbert modular surface. Furthermore we study the condition under which the dimension of the Hilbert modular forms of weight 1/2 or 3/2 be obtained. However we need further investigation about this.
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