We make use of Hecke operators and arithmetic of imaginary quadratic fields to derive an explicit version of a special case of Siegel's mass formula.

A very special case of Siegel's mass formula and Hecke operators

The recent development of the theory of modular forms and associated zeta functions, together with all its arithmetic significance, is quite pleasing, and our knowledge in this field is evergrowing, but the forms of half integral weight have attracted only casual attention, in spite of their importance and ancientness. Indeed, the connection of such forms with zeta functions was never clarified. When Hecke developed his theory of Euler product forthe forms of integral weight, he pointed out the impossibility of a similar theory for the forms of half integral weight, and that only partial information could be obtained for the Fourier coefficients of such forms (Werke, p. 639). He explained this in more detail in his last paper [3], which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases. A treatment of a more general type of modular form was given by Wohlfahrt [12]. In fact, he defined Hecke operators whose degree is the square of a prime, and showed a certain multiplicative relation, as predicted by Hecke, for the Fourier coefficients, but discussed neither Euler product, nor connection with zeta functions. In the present paper, we try to reveal a more affirmative aspect of the subject. To be specific, put, for each positive integer N,

On Modular Forms of Half Integral Weight

Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves as Algebraic Curves.- The Eichler-Shimura Relation and L-functions.- Galois Representations.

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A First Course in Modular Forms

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

Uber Die Analytische Theorie Der Quadratischen Formen III

I. Classical Theory.- I. Modular Forms.- 1. The Modular Group.- 2. Modular Forms.- 3. The Modular Function j.- 4. Estimates for Cusp Forms.- 5. The Mellin Transform.- II. Hecke Operators.- 1. Definitions and Basic Relations.- 2. Euler Products.- III. Petersson Scalar Product.- 1. The Riemann Surface ?\?.- 2. Congruence Subgroups.- 3. Differential Forms and Modular Forms.- 4. The Petersson Scalar Product.- Appendix by D. Zagier. The Eichler-Selberg Trace Formula on SL2(Z).- II. Periods of Cusp Forms.- IV. Modular Symbols.- 1. Basic Properties.- 2. The Manin-Drinfeld Theorem.- 3. Hecke Operators and Distributions.- V. Coefficients and Periods of Cusp Forms on SL2(Z).- 1. The Periods and Their Integral Relations.- 2. The Manin Relations.- 3. Action of the Hecke Operators on the Periods.- 4. The Homogeneity Theorem.- VI. The Eichler-Shimura Isomorphism on SL2(Z).- 1. The Polynomial Representation.- 2. The Shimura Product on Differential Forms.- 3. The Image of the Period Mapping.- 4. Computation of Dimensions.- 5. The Map into Cohomology.- III. Modular Forms for Congruence Subgroups.- VII. Higher Levels.- 1. The Modular Set and Modular Forms.- 2. Hecke Operators.- 3. Hecke Operators on q-Expansions.- 4. The Matrix Operation.- 5. Petersson Product.- 6. The Involution.- VIII. Atkin-Lehner Theory.- 1. Changing Levels.- 2. Characterization of Primitive Forms.- 3. The Structure Theorem.- 4. Proof of the Main Theorem.- IX. The Dedekind Formalism.- 1. The Transformation Formalism.- 2. Evaluation of the Dedekind Symbol.- IV. Congruence Properties and Galois Representations.- X. Congruences and Reduction mod p.- 1. Kummer Congruences.- 2. Von Staudt Congruences.- 3. q-Expansions.- 4. Modular Forms over Z[1/2, 1/3].- 5. Derivatives of Modular Forms.- 6. Reduction mod p.- 7. Modular Forms mod p, p?5.- 8. The Operation of ? on M?.- XI. Galois Representations.- 1. Simplicity.- 2. Subgroups of GL2.- 3. Applications to Congruences of the Trace of Frobenius.- Appendix by Walter Feit. Exceptional Subgroups of GL2.- V. p-Adic Distributions.- XII. General Distributions.- 1. Definitions.- 2. Averaging Operators.- 3. The Iwasawa Algebra.- 4. Weierstrass Preparation Theorem.- 5. Modules over Zp[[T]].- XIII. Bernoulli Numbers and Polynomials.- 1. Bernoulli Numbers and Polynomials.- 2. The Integral Distribution.- 3. L-Functions and Bernoulli Numbers.- XIV. The Complex L-Functions.- 1. The Hurwitz Zeta Function.- 2. Functional Equation.- XV. The Hecke-Eisenstein and Klein Forms.- 1. Forms of Weight 1.- 2. The Klein Forms.- 3. Forms of Weight 2.

A very special case of Siegel's mass formula and Hecke operators

References

On Modular Forms of Half Integral Weight

A First Course in Modular Forms

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

Uber Die Analytische Theorie Der Quadratischen Formen III

Introduction to Modular Forms

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