Abstract: Modular forms of half-integral weight are of intrinsic interest: many of the functions of classical number theory transform under a matrix group with half-integral weight. The aim of this paper is to refine some results and techniques which have been introduced to study these functions and the arithmetic information which they contain.

Abstract: The outline of the paper is s follows: In Sect. l we recall the addition and the transformation formulas of theta functions, which are studied in [3], [4]. We shall in Sect. 2 give a criterion (Th. 1) of irreducibility for a point of Sr Using this criterion, our main theorem (Th. 2) stated above will be proved in Sect. 3. In the last section 4, we shall give a remark on the projective embedding of Kummer varieties s another application of the criterion of irreducibility.

Abstract: In the theory of integral quadratic forms or in the allied subject of modular forms very little seems to be known about the linear independence of theta series associated to positive definite forms. For a system of inequivalent positive binary quadratic forms with a fixed fundamental discriminant, Hecke [4-1 showed that their theta series are independent. However, for positive forms in three or more variables, even in a fixed genus, there does not appear to be any non-trivial general result of a similar nature in that certain explicitly described forms in the genus yield linearly independent theta series. We consider here even positive definite ternary quadratic forms of discriminant 2p and quaternary quadratic forms of discriminant p where p is a prime congruent to 1 mod4. For a fixed such p these forms belong to a single genus G'.=G(3,2p), resp. G(4,p). Let G' denote the subset consisting of those forms in G which represent 2. In [6] Kitaoka observed that the number of form-classes in G'(4, p)

Abstract: An elliptic modular form of positive integral weight k is a linear combination of a cusp form and Eisenstein series. This well-known fact was proved by Hecke and was generalized by Kloostermann to the Hilbert modular case (Hecke [5] and Kloostermann [8]). The case k = 1, which was left unsettled in [-8], was solved by Gundlach under certain conditions; a Hilbert modular form for the full modular group over a real quadratic field is a linear combination of a cusp form and Eisenstein series (Gundlach [3, 4]). The aim of the present paper is to show that the same holds in general for any principal congruence subgroup over a totally real number field. As one expects by [3], the number of linearly independent Eisenstein series of weight 1 is less than (roughly equal to one half of) the number of cusps. It is enough to prove that the number of linearly independent non-cuspidal forms equals that of linearly independent Eisenstein series. To do this, we shall apply the theory of automorphic representations (the equivalence condition, the uniqueness of Whittaker models, etc .... ) as is developed in Jacquet and Langlands [6].

Abstract: In this paper a theory of zeta-functions with Euler product and functional equation is constructed for the case of Hilbert-Siegel functions of a totally real one-class field of algebraic numbers. Bibliography: 14 titles

Abstract: This note determines the quadratic field generated by the square root of the discriminant of the modular equation satisfied by the special value j(α) of the modular function α for a an integer in an imaginary quadratic field.

Abstract: where H 2 is the Siegel upper half space of degree 2 and r2(2) = ker{Sp(4,~)~Sp(4,~/2~)} . The quotient P2(2)\H2 is a non-compact complex manifold. It can be compactified in a minimal way (Satake compactification) to a singular algebraic variety F2(2)\H ~ by adding 15 copies of F(2)\H and 15 points such that each of these points occurs as a cusp of three copies of F(2)\H . In this way we add 15 projective lines F(2)\H* forming a configuration which is explained by the modular interpretation of r2(2)\H ~ .

Abstract: We begin this chapter with an exposition of a theory of theta functions. Using the classical transformation properties of theta functions and the theta function identity (12.7.13), we show that the string functions are modular forms and find a transformation law for these forms. Furthermore, using the theory of modular forms, we prove the “very strange” formula (12.3.7), which in turn, is used to show that the string functions, multiplied by a “standard” cusp-form, are cusp-forms. All this is applied to find explicit formulas for the weight multiplicities and characters of integrable highest weight modules.

Abstract: In [1], [2], Andrianov constructed a remarkable Hecke theory for Siegel's modular forms of degree two. In this article we extend some of his results to the case of vector valued Siegel's modular forms of degree two.

Abstract: a(p) = 2p~ @ ) cos 0(p). Since we know the truth of the Ramanujan-Petersson conjecture, it follows that the 0(p)'s are real. Inspired by the Sato-Tate conjecture for elliptic curves, Serre [14] conjectured that the 0(p)'s are uniformly distributed in the interval [0, rc] with respect to the 1 measure -sin2OdO. Following Serre, we shall refer to this as the Sato-Tate r~ conjecture, there being no room for confusion. It has been known for a long time that the truth of this conjecture implies much about the oscillatory behaviour of the Fourier coefficients. In particular, the following is implied by the Sato-Tate conjecture. Theorem 1. For any normalized Hecke eigenform,