Modular Forms Research Articles

Some congruences for 12-coloured generalized Frobenius partitions

AbstractIn his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$. In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$.

Some new results on generalized Hyers-Ulam stability in modular function spaces

Some new results on generalized Hyers-Ulam stability in modular function spaces

Topology-Based Modularity and Modularity Density for Detecting Protein Complexes: A Comparative Study

Binary relations or interactions among bio-entities, such as proteins, set up the essential part of any living biological system. Protein-protein interactions are usually structured in a graph data structure called "protein-protein interaction networks" (PPINs). Analysis of PPINs into complexes tries to lay out the significant knowledge needed to answer many unresolved questions, including how cells are organized and how proteins work. However, complex detection problems fall under the category of non-deterministic polynomial-time hard (NP-Hard) problems due to their computational complexity. To accommodate such combinatorial explosions, evolutionary algorithms (EAs) are proven effective alternatives to heuristics in solving NP-hard problems. The main aim of this study is to make a close examination of the performance of the EAs where modularity and modularity density are selected as two different objective functions. Topology-based modularity and topology-based modularity density are designed to examine the detection ability of the EAs and to compare their performance. To conduct the experiments, two yeast Saccharomyces cerevisiae PPINs are used and evaluated under nine evaluation metrics. The results reveal the potential impact of the topology-based modularity density to outperform the counterpart modularity functions in almost all evaluation metrics.

Explicit evaluation of triple convolution sums of the divisor functions

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums [Formula: see text] for odd integers [Formula: see text] and [Formula: see text], where [Formula: see text] is the sum of the [Formula: see text]th powers of the positive divisors of [Formula: see text]. We consider four cases, namely (i) [Formula: see text], (ii) [Formula: see text]; [Formula: see text] (iii) [Formula: see text]; [Formula: see text] and (iv) [Formula: see text], and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer [Formula: see text] by certain positive definite quadratic forms. The existing formulas for [Formula: see text] (in [20]), [Formula: see text] (in [7]), [Formula: see text] (in [35]), [Formula: see text], [Formula: see text] (in [30]) and [Formula: see text] (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

Higher d Eisenstein series and a duality-invariant distance measure

The Petersson inner product is a natural inner product on the space of modular invariant functions. We derive a formula, written as a convergent sum over elementary functions, for the inner product Es(G, B) of the real analytic Eisenstein series \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${E}_{s}\\left(\ au ,\\overline{\ au }\\right)$$\\end{document} and a general point in Narain moduli space. We also discuss the utility of the Petersson inner product as a distance measure on the space of 2d CFTs, and apply our procedure to evaluate this distance in various examples.

WILES DEFECT OF HECKE ALGEBRAS VIA LOCAL-GLOBAL ARGUMENTS

Abstract In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$ ). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections. We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$ : It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$ . In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$ , and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

Structure of the space of polyharmonic Maass forms with an application to L-values

A polyharmonic Maass form introduced by Lagarias and Rhoades is a generalization of a harmonic Maass form in a sense. In this paper, we construct some bases for the spaces of polyharmonic Maass forms from the structure of the spaces of weakly holomorphic modular forms for prime levels or square-free levels. Furthermore, we relate the polyharmonic Maass forms of depth 1 or depth 3/2 to the L-series of holomorphic cusp forms for square-free levels.

On the modulo p zeros of modular forms congruent to theta series

For a prime p larger than 7, the Eisenstein series of weight p−1 has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval [0,1728]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight k≥4 for the full modular group as the modular forms for which the first dim⁡(Mk) Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.

Comparison of integral structures on the space of modular forms of full level N

Let N≥3 and r≥1 be integers and p≥2 be a prime such that p∤N. One can consider two different integral structures on the space of modular forms over Q, one coming from arithmetic via q-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply methods due to Brian Conrad to the situation of modular forms of even weight and level Γ(Npr) over Qp(ζNpr) to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level pr whenever pr>3, allowing us to compute a lower bound which agrees with the upper bound. Hence we compute the exponent precisely.

Congruences and the Iwasawa Main Conjecture for modular forms

Congruences and the Iwasawa Main Conjecture for modular forms

Twisted Kronecker series and periods of modular forms on Γ0(N)

We introduce an infinite family of Kronecker series twisted by characters. As an application, we give a closed formula for the sum of all Hecke eigenforms on Γ0(N) multiplied by their twisted period polynomials in terms of the product of those twisted Kronecker series, when N is square free. This extends an identity of Zagier among period polynomials, Hecke eigenforms and a quotient of Jacobi theta series.

Congruence modules in higher codimension and zeta lines in Galois cohomology

This article builds on recent work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main results are a criterion for detecting regularity of local rings in terms of congruence modules, and a more refined version of a result tracking the change of congruence modules under deformation. Number theoretic applications include the construction of canonical lines in certain Galois cohomology groups arising from adjoint motives of Hilbert modular forms.

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.

Threonine phosphorylation of STAT1 restricts interferon signaling and promotes innate inflammatory responses

Since its discovery over three decades ago, signal transducer and activator of transcription 1 (STAT1) has been extensively studied as a central mediator for interferons (IFNs) signaling and antiviral defense. Here, using genetic and biochemical assays, we unveil Thr748 as a conserved IFN-independent phosphorylation switch in Stat1, which restricts IFN signaling and promotes innate inflammatory responses following the recognition of the bacterial-derived toxin lipopolysaccharide (LPS). Genetically engineered mice expressing phospho-deficient threonine748-to-alanine (T748A) mutant Stat1 are resistant to LPS-induced lethality. Of note, T748A mice exhibited undisturbed IFN signaling, as well as total expression of Stat1. Further, the T748A point mutation of Stat1 recapitulates the safeguard effect of the genetic ablation of Stat1 following LPS-induced lethality, indicating that the Thr748 phosphorylation contributes inflammatory functionalities of Stat1. Mechanistically, LPS-induced Toll-like receptor 4 endocytosis activates a cell-intrinsic IκB kinase-mediated Thr748 phosphorylation of Stat1, which promotes macrophage inflammatory response while restricting the IFN and anti-inflammatory responses. Depletion of macrophages restores the sensitivity of the T748A mice to LPS-induced lethality. Together, our study indicates a phosphorylation-dependent modular functionality of Stat1 in innate immune responses: IFN phospho-tyrosine dependent and inflammatory phospho-threonine dependent. Better understanding of the Thr748 phosphorylation of Stat1 may uncover advanced pharmacologically targetable molecules and offer better treatment modalities for sepsis, a disease that claims millions of lives annually.

Organic 6G networks: ultra-flexibility through extensive stateless functional split

With the increase in hardware performance, the 5G mobile network architecture shifted from physical components to software-only micro-services. The very modular network functions can be deployed flexibly on commodity hardware. However, the extensive modularity of these network functions is increasing the number of managed entities, and the core network request latency. Also, it requires extensive procedures to be able to re-select the components for specific devices, a fundamental condition for a potential system scale-down. In this paper, we propose a new organic 6G network architecture that handles these challenges through a new functionality split based on the experience of IT software services. Furthermore, we provide an analysis based on main 5G procedures, showing that the newly proposed architecture is handling the re-selection of functionality significantly better, which is a cornerstone of high-speed scaling (especially scaling-out), as well as migration of functionality and users.

Reinforced ECC-UHPC-SCC composite modular framed shear wall systems under lateral cyclic loading

The shear strength and deformability of high performance reinforced composite framed shear wall systems made of Engineered Cementitious Composite (ECC), Ultra-High-Performance Concrete (UHPC) and Self-Consolidating Concrete (SCC) subjected to lateral cyclic shear loading to failure were investigated. A total of seven framed shear wall specimens were constructed using combinations of ECC or SCC wall, SCC frame and ECC or UHPC frame joints. The performance of such novel modular forms of walling systems were compared based on experimental results in terms of cracking, failure mode, hysteretic shear load-deformation response, shear load capacity, ductility, bearing capacity degradation, stiffness degradation, dissipated energy, viscous damping capacity, induced wall shear stresses and strain development characteristics. ECC walling system exhibited better structural performance compared to their SCC counterparts showing up to 40% higher load capacity forming significantly higher number of closely spaced diagonal cracks with up to 72% lower crack shear stress, 14% to 21% higher ductility, up to 2 times higher energy dissipation capacity and lower rate of stiffness/bearing capacity degradation through both hysteretic and damping actions. The incorporation of ECC and ECC or UHPC joint in frame significantly improved shear strength capacity of infill wall and overall framed shear wall system. ECC incorporated framed shear walls even with frames made of SCC and UHPC joints also exhibited superior performance. This study confirmed the superior performance of ECC material for the construction of reinforced framed shear wall with boundary frame made of ECC/SCC/UHPC combinations for better seismic resistance.

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.

Congruence classes for modular forms over small sets

Serre showed that for any integer [Formula: see text] for almost all [Formula: see text] where [Formula: see text] is the [Formula: see text] Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study [Formula: see text] over the set of integers with [Formula: see text] many prime factors. Moreover, we show that any residue class [Formula: see text] can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of [Formula: see text]

Bianchi period polynomials: Hecke action and congruences

Let Γ=PSL2(O)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma ={{\\,\\mathrm{\ extrm{PSL}}\\,}}_2(\\mathcal {O})$$\\end{document} be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight k period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is “dual” to the space of weight k modular symbols for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, reflecting the duality between the first and second cohomology groups of the arithmetic group Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. Using this result, we describe the action of Hecke operators on the space of period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} via the Heilbronn matrices. As in the classical case, spaces of Bianchi period polynomials are related to parabolic cohomology of Bianchi groups, and in turn, via the Eichler–Shimura–Harder Isomorphism, to spaces of Bianchi modular forms. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the above mentioned Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.

Group ring valued Hilbert modular forms

Group ring valued Hilbert modular forms