A note on Siegel modular forms

This paper gives a new identification for Siegel modular forms with respect to any congru-ence subgroup by investigating the properties of their Fourier-Jacobi expansions, and verifies a com-parison theorem for the dimensions of the spaces Snk(n) and Jk,10(n) with small weight k. These results can be used to estimate the dimension of the space of modular forms.

Theta Function and Hilbert Modular Forms of Half Integral Weight

In this paper a kind of theta function is constructed by means of spherical function. And we also obtain some Hilbert modular forms of half integral weight.

p-adic Modular Symbols and ∧-adic Modular Forms

A construction of A-adic modular forms from p-adic modular symbols is described. It shows that each A linear map satisfying some certain conditions from the module of p-adic modular symbols to A corresponds to a A-adic modular form.

Representations by Certain Sextenary Quadratic Forms Whose Coefficients Are 1, 2, 3 and 6

Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.

Generalized Theta-Functions with Characteristics and Cusp Form Corresponding to Quadratic Forms in Ten Variables

The modular properties of generalized theta-functions with characteristics are used to build cusp form corresponding to quadratic forms in ten variables.

Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

Elliptic curves and positive definite ternary forms

For two given ternary quadratic forms f( x, y, z) and g( x, y, z), let r( f, n) and r( g,n) be the numbers of representations of n represented by f( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we have r( f, n) = r( g, n) or r( f, n) ≠ r( g, n).Our method is to use elliptic curves and the corresponding new forms.

Behavior of Theta-series under Modular Substitutions

In this article we provide a transformation formula of certain theta series, and apply it to obtain non-holomorphic vector-valued modular forms.

Representation of Integers by Ternary Quadratic Forms

In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.

Anomaly cancellation and modularity, II:The E_8× E_8 case

In this paper we show that both of the Green-Schwarz anomaly factorization formula for the gauge group E8 × E8 and the Ho^ava-Witten anomaly factorization formula for the gauge group E8 can be derived through modular forms of weight 14. This answers a question of Schwarz. We also establish generalizations of these factorization formulas and obtain a new Horava-Witten type factorization formula.

Estimates for Fourier Coefficients of Cusp Forms in Weight Aspect

Let f be a holomorphic Hecke eigenform of weight k for the modular groupΓ = SL2(Z) and let λf(n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate 1λf(n21) x2 k2(log(x + k))6n≤x is established, which improves the previous result.

Difference of a Hauptmodul for Γ_(0)(N ) and certain Gross-Zagier type CM value formulas

In this work, we show that the difference of a Hauptmodul for a genus zero group Γ_(0)(N) as a modular function on Y_(0)(N) × Y_(0)(N) is a Borcherds lift of type(2, 2). As applications, we derive the monster denominator formula like product expansions for these modular functions and certain Gross-Zagier type CM value formulas.

Rankin-Cohen Deformations and Representation Theory

The author uses the unitary representation theory of SL2(R) to understand the Rankin-Cohen brackets for modular forms. Then this interpretation is used to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established.

Le Système d’Euler de Kato en Famillie(Ⅱ)

This article is the second article on the generalization of Kato’s Euler system.The main subject of this article is to construct a family of Kato’s Euler systems over the cuspidal eigencurve,which interpolate the Kato’s Euler systems associated to the modular forms parametrized by the cuspidal eigencurve.We also explain how to use this family of Kato’s Euler system to construct a family of distributions on Z_p over the cuspidal eigencurve;this distribution gives us a two-variable p-adic L function which interpolate the p-adic L function of modular forms.

Ramanujan-type congruences for broken 2-diamond partitions modulo 3

The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).