On the Petersson Norm of Certain Siegel Modular Forms

We shall prove a rationality result for a quotient of scalar products involving the Ikeda lift of an elliptic cusp form.     

Linear relations of theta series of genera of quadratic forms

In this paper we study linear relations among theta series of genera of positive definite n-ary quadratic forms with given level D,2D,4D and 8D for square free D. We obtain a basis for the space generated by genus theta series. This forms a basis of Eisenstein space.     

A short remark on sign changes of Fourier coefficients of half-integral weight modular forms

We give an upper bound on the first sign change of the Fourier coefficients of a cusp form of half-integral weight.     

Parabolic cohomology and multiple Hecke L-values

The Ramanujan Journal - We derive various identities among the special values of multiple Hecke L-series. We show that linear combinations of multiple Hecke L-values can be expressed as linear...     

Weight-dependent congruence properties of modular forms

In this paper, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions, in terms of the weights of modular forms, for a modular form to be non-p-ordinary. As applications of our main theorem we derive a linear relation among coefficients of new forms. Furthermore, congruence relations among special values of Dedekind zeta functions of real quadratic fields are derived.     

The joint weight enumerators and Siegel modular forms

The weight enumerator of a binary doubly even self-dual code is an isobaric polynomial in the two generators of the ring of invariants of a certain group of order 192. The aim of this note is to study the ring of coefficients of that polynomial, both for standard and joint weight enumerators.

     

Isomorphism classes of elliptic and hyperelliptic curves over finite fields

We give the number and representatives of isomorphism classes of hyperelliptic curves of genus g defined over finite fields , g=1,2,3. These results have applications to hyperelliptic curve cryptography.     

The heat operator and mock Jacobi forms

We study a necessary and sufficient condition for Jacobi integrals of weight -r+\fracj2-r+\frac{j}{2}, r∈ℤ≥0, and index ℳ^(j) on ℋ×ℂ ^j to have a dual Jacobi form of weight r+\fracj2+2r+\frac{j}{2}+2 and index ℳ^(j). Such a meromorphic Jacobi integral with a dual Jacobi form is called a mock Jacobi form whose concept was first introduced by Zagier in Séminaire Bourbaki, 60éme année, 2006–2007, N° 986. In fact, we show the map L^r+1_M^(j)L^{r+1}_{\mathcal{M}^{(j)}} from the space of mock Jacobi forms to that of Jacobi forms is surjective by constructing the corresponding inverse image via Eichler integral of vector valued modular forms which are coming from the theta decomposition of Jacobi forms. We discuss Lerch sums as a typical example.     

<Emphasis Type="Italic">p</Emphasis>-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight

Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL ₂(ℤ) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ₀(4N) for N = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications to our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.