Siegel Modular Forms Research Articles

On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms

On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms

Theta blocks

We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta function and show that they give a new powerful method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers to this question.

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.

Sp(6, Z) modular symmetry in flavor structures: quark flavor models and Siegel modular forms for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\\Delta }\\left(96\\right)$$\\end{document}

We study an approach to construct Siegel modular forms from Sp(6, Z). Zero-mode wave functions on T6 with magnetic flux background behave Siegel modular forms at the origin. Then T-symmetries partially break depending on the form of background magnetic flux. We study the background such that three T-symmetries TI, TII and TIII as well as the S-symmetry remain. Consequently, we obtain Siegel modular forms with three moduli parameters (ω1, ω2, ω3), which are multiplets of finite modular groups. We show several examples. As one of examples, we study Siegel modular forms for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\\Delta }\\left(96\\right)$$\\end{document} in detail. Then, as a phenomenological applicantion, we study quark flavor models using Siegel modular forms for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\\Delta }\\left(96\\right)$$\\end{document}. Around the cusp, ω1 = i∞, the Siegel modular forms have hierarchical values depending on their TI-charges. We show the deviation of ω1 from the cusp can generate large quark mass hierarchies without fine-tuning. Furthermore CP violation is induced by deviation of ω2 from imaginary axis.

Multiplier systems for Siegel modular groups

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus [Formula: see text] must be integral or half integral. Actually he proved that for a system [Formula: see text] of complex numbers of absolute value 1 [Formula: see text] can be an automorphy factor only if [Formula: see text] is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for [Formula: see text] and [Formula: see text], Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.

Modular symmetry in magnetized T2g torus and orbifold models

We study the modular symmetry in magnetized T2g torus and orbifold models. The T2g torus has the modular symmetry Γg=Sp(2g,Z). The magnetic flux background breaks the modular symmetry to a certain normalizer Ng(H). We classify remaining modular symmetries by magnetic flux matrix types. Furthermore, we study the modular symmetry for wave functions on the magnetized T2g and certain orbifolds. It is found that wave functions on magnetized T2g as well as its orbifolds behave as the Siegel modular forms of weight 1/2 and N˜g(H,h), which is the metaplectic congruence subgroup of the double covering group of Ng(H), N˜g(H). Then, wave functions transform nontrivially under the quotient group, N˜g,h=N˜g(H)/N˜g(H,h), where the level h is related to the determinant of the magnetic flux matrix. Accordingly, the corresponding four-dimensional chiral fields also transform nontrivially under N˜g,h modular flavor transformation with modular weight −1/2. We also study concrete modular flavor symmetries of wave functions on magnetized T2g orbifolds. Published by the American Physical Society 2024

Rademacher Expansion of a Siegel Modular Form for N=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {N}}}= 4$$\\end{document} Counting

The degeneracies of 1/4 BPS states with unit torsion in heterotic string theory compactified on a six torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form Φ10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi _{10}$$\\end{document} of weight 10. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher-type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of 1/Φ10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\Phi _{10}$$\\end{document}. The construction uses two distinct SL(2,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SL}(2, {\\mathbb {Z}})$$\\end{document} subgroups of Sp(2,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Sp}(2, {\\mathbb {Z}})$$\\end{document} which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of 1/η24\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\eta ^{24}$$\\end{document} by means of a continued fraction structure.

Universally irreducible subvarieties of Siegel moduli spaces

Abstract A subvariety of a quasi-projective complex variety X is called “universally irreducible” if its preimage inside the universal cover of X is irreducible. In this paper we investigate sufficient conditions for universal irreducibility. We consider in detail complete intersection subvarieties of small codimension inside Siegel moduli spaces of any finite level. Moreover, we show that, for g ≥ 3 {g\geq 3} , every Siegel modular form is the product of finitely many irreducible analytic functions on the Siegel upper half-space ℍ g {{\mathbb{H}}_{g}} . We also discuss the special case of singular theta series of weight 1 2 {\frac{1}{2}} and of Schottky forms.

Cuspidal components of Siegel modular forms for large discrete series representations of $$\textrm{Sp}_4({\mathbb {R}})$$

Cuspidal components of Siegel modular forms for large discrete series representations of $$\textrm{Sp}_4({\mathbb {R}})$$

Differentiating Siegel Modular Forms and the Moving Slope of 𝒜g

Abstract We study the cone of moving divisors on the moduli space ${\mathcal{A}}_{g}$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on ${\mathcal{A}}_{g}$ for $g\leq 4$, and gives an explicit upper bound for the moving slope of ${\mathcal{A}}_{5}$ and a conjectural upper bound for the moving slope of ${\mathcal{A}}_{6}$.

Siegel theta series for quadratic forms of signature (m − 1,1)

We investigate Siegel theta series for quadratic forms of signature (m−1,1). On the one hand, we construct a holomorphic series that does not transform like a modular form. On the other hand, we construct a non-holomorphic series that transforms like a Siegel modular form of weight m/2. Moreover, the holomorphic series describes almost everywhere the holomorphic part of the modular series.

On p-divisibility of Fourier coefficients of Siegel modular forms

On p-divisibility of Fourier coefficients of Siegel modular forms

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 p-adic Siegel Eisenstein series

A generalization of Serre's p-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a p-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is proved.

Eigenform product identities of genus two Siegel modular forms of general congruence level

Given two eigenforms, it is a natural question to ask if the product of the eigenforms is again an eigenform. In the case of elliptic modular forms, this was answered in the full level case by Duke and Ghate and in the general level case by Johnson. In this paper, we consider the case of genus two Siegel modular forms with general congruence level.

Sl(2)ˆ Decomposition of denominator formulae of some BKM Lie superalgebras - II

The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did not appear in that case. We restrict our analysis to the first N terms in the expansion as a first attempt at deconstructing the Siegel modular forms and unravelling the structure of potentially new Lie algebras that occur for N=5,6.

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.

On the classification of $$({\mathfrak {g}},K)$$-modules generated by nearly holomorphic Hilbert–Siegel modular forms and projection operators

We classify the \(({\mathfrak {g}},K)\)-modules generated by nearly holomorphic Hilbert–Siegel modular forms by the global method. As an application, we study the image of projection operators on the space of nearly holomorphic Hilbert–Siegel modular forms with respect to infinitesimal characters in terms of \(({\mathfrak {g}},K)\)-modules.

Positivity of discrete information for CHL black holes

Black holes carry more information about the microstates than just the total degeneracy. As a concrete example, the ZN-twined helicity trace indices for 14-BPS black holes of the CHL models allow extracting information about the distribution of the ZN charges among the black hole microstates. The number of black hole microstates carrying a definite eigenvalue under the generator of the ZN twining group must be positive. This leads to a specific prediction for the signs of certain linear combinations of Fourier coefficients of Siegel modular forms. We explicitly test these predictions for low charges. In the D1-D5-P duality frame, we compute the appropriate hair removed partition functions and show the positivity of the appropriate Fourier coefficients for low charges. We present various consistency checks on our computations.

Siegel Eisenstein series of level two and its applications

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number [Formula: see text] associated with the maximal order in the quaternion algebra [Formula: see text]. As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun’s “1-3-5 conjecture”. Furthermore, we show that “a perfect square” in the integral condition version of Sun’s conjecture can be replaced by “a power of 4”.