Siegel Modular Forms Of Degree Research Articles

Siegel modular forms of degree three and invariants of ternary quartics

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4 4 , 12 12 , 12 12 , 14 14 , 18 18 , 20 20 and 30 30 . We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators.

Dimension formulas for Siegel modular forms of level 4

AbstractWe prove several dimension formulas for spaces of scalar‐valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of whose local component at admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at , we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.

On Hermitian Eisenstein series of degree $2$

We consider the Hermitian Eisenstein series $E^{(\mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $\mathbb{K}$ and determine the influence of $\mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(\mathbb{K})}^2}_4 = E^{{(\mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $\mathbb{K} = \mathbb{Q}(\sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(\mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $\mathbb{K}\neq \mathbb{Q}(\sqrt{-3})$. The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maa{\ss} space that does not vanish identically for sufficiently large weight; however, when the weight is fixed, we will see that it tends to $0$ as the discriminant tends to $-\infty$. Finally, we show that these forms generate the space of cusp forms in the Maaß Spezialschar as a module over the Hecke algebra as $\K$ varies over imaginary-quadratic number fields.

Siegel modular forms of degree two and level five

We construct a ring of meromorphic Siegel modular forms of degree 2 and level 5, with singularities supported on an arrangement of Humbert surfaces, which is generated by four singular theta lifts of weights 1, 1, 2, 2 and their Jacobian. We use this to prove that the ring of holomorphic Siegel modular forms of degree 2 and level Gamma _0(5) is minimally generated by eighteen modular forms of weights 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 10, 11, 11, 11, 13, 13, 13, 15.

Lowest weight modules of Sp4(R) and nearly holomorphic Siegel modular forms

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight $det^\ell$ sym^m with respect to an arbitrary congruence subgroup of Sp_4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight $det^{\ell'} sym^{m'}$ with $(\ell', m')$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight $det^{3}sym^{m'}$ that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.

Nearly holomorphic automorphic forms on SL2

We define the space of nearly holomorphic automorphic forms on a connected reductive group G over Q such that the homogeneous space G(R)1/K∞∘ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's study, there are the classification of indecomposable (g,K∞)-modules which occur in the space of nearly holomorphic elliptic modular forms and Siegel modular forms of degree 2. This paper studies global representations of the adele group G(AQ) which occur in the space of nearly holomorphic Hilbert modular forms. In the case of elliptic modular forms, the result of this paper is an adelization of Pitale, Saha and Schmidt's result.

The ring of modular forms of degree two in characteristic three

We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3.

On the first Fourier-Jacobi coefficient of Siegel modular forms of degree two

In this paper we prove that the first Fourier-Jacobi coefficient of a non-zero Siegel cusp form and a Hecke eigenform of degree 2, weight k, for Sp4(Z) is not identically zero.

Maass relations for Saito\u2013Kurokawa lifts of higher levels

It is known that among Siegel modular forms of degree 2 and level 1 the only functions that violate the Ramanujan conjecture are Saito–Kurokawa lifts of modular forms of level 1. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for mathrm {GSp}_4 is predicted to fail only in case of CAP representations. It is not known though whether the associated Siegel modular forms (of various levels) still satisfy a version of Maass relations. We show that this is indeed the case for the ones related to P-CAP representations. Our method generalises an approach of Pitale, Saha and Schmidt who employed representation–theoretic techniques to (re)prove this statement in case of level 1. In particular, we compute and express certain values of a global Bessel period in terms of Fourier coefficients of the associated Siegel modular form. Moreover, we derive a local–global relation satisfied by Bessel periods, which allows us to combine those computations with a characterization of local components of CAP representations.

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of $L$-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.

Constructions of nearly holomorphic Siegel modular forms of degree two

Pitale, Saha and Schmidt studied the representation theoretic aspects of nearly holomorphic modular forms. By their theory, we obtain a classification of [Formula: see text]-modules which occur in the space of nearly holomorphic modular forms. In this paper, we give two constructions of nearly holomorphic Siegel modular forms of degree [Formula: see text] which generate reducible indecomposable modules. One construction is given by the Rankin–Cohen bracket of Shimura’s Eisenstein series and the other by Klingen Eisenstein series.

Some comments on symmetric orbifolds of K3

We consider two dimensional mathcal{N} = (4, 4) superconformal field theories in the moduli space of symmetric orbifolds of K3. We complete a classification of the discrete groups of symmetries of these models, conditional to a series of assumptions and with certain restrictions. Furthermore, we provide a partial classification of the set of twining genera, encoding the action of a discrete symmetry g on a space of supersymmetric states in these models. These results suggest the existence of a number of surprising identities between seemingly different Borcherds products, representing Siegel modular forms of degree two and level N > 1. We also provide a critical review of various properties of the moduli space of these superconformal field theories, including the groups of dualities, the set of singular models and the locus of symmetric orbifold points, and describe some puzzles related to our (lack of) understanding of these properties.

Correction to: On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

Correction to: On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

Covariants of binary sextics and modular forms of degree 2 with character

We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree $2$ with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular form defined by a covariant we express the order of vanishing along the locus of products of elliptic curves in terms of the covariant.

Sturm bounds for Siegel modular forms of degree 2 and odd weights

We correct the proof of the theorem in the previous paper presented by Kikuta, which concerns Sturm bounds for Siegel modular forms of degree 2 and of even weights modulo a prime number dividing $$2\cdot 3$$ . We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.

Petersson norms of not necessarily cuspidal Jacobi modular forms and applications

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain “growth-killing” invariant differential operators on H2 and those on H1×C (here Hn denotes the Siegel upper half space of degree n). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms; and as a by-product we also show that any Siegel modular form of degree 2 is determined by its ‘fundamental’ Fourier coefficients.

On Vector-Valued Siegel Modular Forms of Degree 2 and Weight $(j,2)$

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight \mathrm{Sym}^j\otimes\mathrm{det}^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.

On vector-valued Siegel modular forms of degree 2 and weight (j,2)

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.

Archimedean aspects of Siegel modular forms of degree 2

We survey the archimedean representations and Langlands parameters corresponding to holomorphic Siegel modular forms of degree~$2$. This leads to a determination of archimedean local factors for various $L$-functions and all vector-valued weights. We determine the Hodge structures that correspond to holomorphic Siegel modular forms and clarify the relationship with four-dimensional symplectic artin representations.

Archimedean aspects of Siegel modular forms of degree 2

We survey the archimedean representations and Langlands parameters corresponding to holomorphic Siegel modular forms of degree~$2$. This leads to a determination of archimedean local factors for various $L$-functions and all vector-valued weights. We determine the Hodge structures that correspond to holomorphic Siegel modular forms and clarify the relationship with four-dimensional symplectic artin representations.