Hermitian Modular Forms Of Degree Research Articles

Graded rings of Hermitian modular forms with singularities

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group mathrm {SU}_{2,2}({mathcal {O}}_K) where K is the imaginary-quadratic number field of discriminant -d, d in {4, 7,8,11,15,19,20,24} we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

Koecher-Maass series associated to Hermitian modular forms of degree 2 and a characterization of cusp forms by the Hecke bound

We establish basic analytic properties of the Koecher-Maass series associated to Hermitian modular forms of degree 2, level N, twisted by Maass forms on 3-dimensional hyperbolic space. As one of its applications, we give a characterization of degree 2 Hermitian cusp forms by the growth of their Fourier coefficients.

Two graded rings of Hermitian modular forms

We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in {mathbb {Q}}(sqrt{-7}) and {mathbb {Q}}(sqrt{-11}). In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms.

Congruences in Hermitian Jacobi and Hermitian modular forms

Abstract In this paper, we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod p theory of Hermitian Jacobi forms over ℚ ⁢ ( i ) {\mathbb{Q}(i)} . We then apply the mod p theory of Hermitian Jacobi forms to characterize U ⁢ ( p ) {U(p)} congruences and to study Ramanujan-type congruences for Hermitian Jacobi forms and Hermitian modular forms of degree 2 over ℚ ⁢ ( i ) {\mathbb{Q}(i)} .

A ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients

We determine the structure over $$\mathbb {Z}$$ of a ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients whose weights are multiples of 4 when the base field is the Gaussian number field $$\mathbb {Q}(\sqrt{-1})$$. Namely, we give a set of generators consisting of 24 modular forms. As an application of our structure theorem, we give the Sturm bounds for such Hermitian modular forms of weight k with $$4\mid k$$, for $$p=2$$, 3. We remark that the bounds for $$p\ge 5$$ are already known.

On the theta operator for Hermitian modular forms of degree 2

The mod p kernel of the theta operator is the set of modular forms whose image of the theta operator is congruent to zero modulo a prime p. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa’s odd weight cusp form is an element of mod 23 kernel of the theta operator. In this paper, we give some examples which represent elements in the mod p kernel of the theta operator in the case of Hermitian modular forms of degree 2.

Koecher–Maass series of the Ikeda lift for U ( m , m )

Let K be an imaginary quadratic field with discriminant -D, and x the Dirichlet character corresponding to the extension K/Q. Let m=2n or 2n+1 with n a positive integer. Let f be a primitive form of weight 2k+1 and level D with Neben character x, or a primitive form of weight 2k for SL(2,Z) according as m=2n, or m=2n+1. For such an f let I(m,f) be the lift of f to the space of Hermitian modular forms of degree m constructed by Ikeda. We then give an explicit formula of the Koecher-Maass series L(s,I(m,f)) of I(m,f). This is a generalization of [Y. Mizuno, Manuscripta Math. 119(2006), 175-216].

Congruences for Hermitian modular forms of degree 2

We give two congruence properties of Hermitian modular forms of degree 2 over Q(−1) and Q(−3). The one is a congruence criterion for Hermitian modular forms which is generalization of Sturmʼs theorem. Another is the well-definedness of the p-adic weight for Hermitian modular forms.

The graded ring of Hermitian modular forms of degree 2 over [formula omitted

We describe the graded ring of symmetric Hermitian modular forms of even weights and degree 2 over Q ( −2 ) in terms of generators and relations. All the 8 generators of weight up to 12 are Maaß lifts and some of them can also be obtained from Borcherds products. Moreover, we construct generators for the module over this ring consisting of all Hermitian modular forms with respect to the commutator subgroup. As an application the field of Hermitian modular functions over Q ( −2 ) is determined. Finally, we construct 5 algebraically independent symmetric Hermitian modular forms in terms of theta series.

The graded ring of Hermitian modular forms of degree 2 over

We describe the graded ring of symmetric Hermitian modular forms of even weights and degree 2 over Q ( −2 ) in terms of generators and relations. All the 8 generators of weight up to 12 are Maaß lifts and some of them can also be obtained from Borcherds products. Moreover, we construct generators for the module over this ring consisting of all Hermitian modular forms with respect to the commutator subgroup. As an application the field of Hermitian modular functions over Q ( −2 ) is determined. Finally, we construct 5 algebraically independent symmetric Hermitian modular forms in terms of theta series.

Graded rings of Hermitian modular forms of degree 2

At first it is shown that any Siegel modular form of degree 2 and even weight can be lifted to a Hermitian modular form of degree 2 over any imaginary-quadratic number field K. In the cases \(\) and \(\) we describe the graded rings of Hermitian modular forms with respect to all abelian characters. Generators are constructed as Maaslifts or as Borcherds products. The description allows a characterization in terms of generators and relations. Moreover we show that the graded rings are generated by theta constants by analyzing the associated five-dimensional representation of PSp2(ℤ/3ℤ). As an application the fields of Hermitian modular functions are determined.

The graded ring of hermitian modular forms of degree 2

The graded ring of hermitian modular forms of degree 2

Differential operators on Hermitian Jacobi forms

We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms.

A note on the structure of the ring of symmetric Hermitian modular forms of degree 2 over the Gaussian field

A note on the structure of the ring of symmetric Hermitian modular forms of degree 2 over the Gaussian field

A Dirichlet series for Hermitian modular forms of degree 2

A Dirichlet series for Hermitian modular forms of degree 2

An arithmetic of Hermitian modular forms of degree two

An arithmetic of Hermitian modular forms of degree two