Search

In this paper, we construct generalized L-functions associated to meromorphic modular forms of weight 1 2 for the theta group with a single simple pole in the fundamental domain. We then consider their behavior toward i∞ and relate this to the Riemann zeta function. [ABSTRACT FROM AUTHOR]

By using modular functions on the upper complex half-plane, we study a class of strain energies for crystalline materials whose global invariance originates from the full symmetry group of the underlying lattice. This follows Ericksen's suggestion which aimed at extending the Landau-type theories to encompass the behavior of crystals undergoing structural phase transformation, with twinning, microstructure formation, and possibly associated plasticity effects. Here we investigate such Ericksen-Landau strain energies for the modelling of reconstructive transformations, focusing on the prototypical case of the square-hexagonal phase change in 2D crystals. We study the bifurcation and valley-floor network of these potentials, and use one in the simulation of a quasi-static shearing test. We observe typical effects associated with the micro-mechanics of phase transformation in crystals, in particular, the bursty progress of the structural phase change, characterized by intermittent stress-relaxation through microstructure formation, mediated, in this reconstructive case, by defect nucleation and movement in the lattice. [ABSTRACT FROM AUTHOR]

Our results investigate mock theta functions and quantum modular forms via quantum q-series identities. After Lovejoy, quantum q-series identities are such that they do not hold as an equality between power series inside the unit disc in the classical sense, but do hold at dense sets of roots of unity on the boundary. We establish several general (multivariable) quantum q-series identities and apply them to various settings involving (universal) mock theta functions. As a consequence, we surprisingly show that limiting, finite, universal mock theta functions at roots of unity for which their infinite counterparts do not converge are quantum modular. Moreover, we show that these finite limiting universal mock theta functions play key roles in (generalized) Ramanujan radial limits. A further corollary of our work reveals that the finite Kontsevich–Zagier series is a kind of "universal quantum mock theta function," in that it may be used to evaluate odd-order Ramanujan mock theta functions at roots of unity. (We also offer a similar result for even-order mock theta functions.) Finally, to complement the notion of a quantum q-series identity and the results of this paper, we also define what we call an "antiquantum q-series identity' and offer motivating general results with applications to third-order mock theta functions. [ABSTRACT FROM AUTHOR]

In this work, we present a new weighted method for proving the generalized Hyers-Ulam stability for nonlinear Volterra integral equations in modular spaces. Using the same technique, we also prove the generalized Hyers-Ulam stability for nonlinear functional equations under Δ2 conditions. Fixed-point theorems in modular spaces form the foundation of our main conclusions. [ABSTRACT FROM AUTHOR]

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system \Phi. We introduce the generating function Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for \Phi. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of \Phi, with automorphic Weyl denominators, we express Z(\mathbf {s}) in terms of Jacobi theta functions and the Siegel modular form \Delta _5. We also show that the domain of convergence of Z(\mathbf {s}) is the Tits cone of \Phi, and discover that this domain inherits the intricate geometric structure of Apollonian packings. [ABSTRACT FROM AUTHOR]

We prove Rogers–Ramanujan-type identities for the Nahm sums associated with the tadpole Cartan matrix of rank 3. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Calinescu, Penn and the first author in this case. We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group. We also present some conjectures for a general rank. [ABSTRACT FROM AUTHOR]

Viazovska's solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska's work is no exception. In this article, we'll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska. [ABSTRACT FROM AUTHOR]

Let p be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form E k ∗ V (E k ∗) where E k ∗ is a classical, normalized Eisenstein series on Γ 0 (p) and V the p-adic Frobenius operator. In particular, we extend our previous paper to the primes 2 and 3. For these primes our main theorem improves somewhat upon earlier results by Emerton, Buzzard and Kilford, and Roe. We include a detailed discussion of those earlier results as seen from our perspective. We also give some improvements to our earlier paper for primes p ≥ 5 . Apart from establishing these improvements, our main purpose here is also to show that all of these results can be obtained by a uniform method, i.e., a method where the main points in the argumentation is the same for all primes. We illustrate the results by some numerical examples. [ABSTRACT FROM AUTHOR]

String compactifications on an orbi-folded torus with complex structure give rise to chiral fermions, spontaneously broken CP, modular invariance. We show that this allows simple effective theories of flavour and CP where: i) the QCD angle vanishes; ii) the CKM phase is large; iii) quark and lepton masses and mixings can be reproduced up to order one coefficients. We implement such general paradigm in supersymmetry or supergravity, with modular forms or functions, with or without heavy colored states. [ABSTRACT FROM AUTHOR]

We use Poincaré series for massive Maass-Jacobi forms to define a "massive theta lift", and apply it to the examples of the constant function and the modular invariant j-function, with the Siegel-Narain theta function as integration kernel. These theta integrals are deformations of known one-loop string threshold corrections. Our massive theta lifts fall off exponentially, so some Rankin-Selberg integrals are finite without Zagier renormalization. [ABSTRACT FROM AUTHOR]

This text is a short and elementary introduction to the 'monstrous moonshine' aiming to be as accessible as possible. We first review the classification of finite simple groups out of which the monster naturally arises and the latter's features that are needed to state the moonshine conjecture of Conway and Norton. Then, we motivate modular functions and forms from the classification of complex tori, with the definitions of the J-invariant and its q-expansion as a goal. We eventually provide evidence for the monstrous moonshine correspondence, state the conjecture, and then introduce the ideas that led to its proof. Lastly, we give a brief account of some recent developments and current research directions in the field. [ABSTRACT FROM AUTHOR]

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, which reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker–Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2. [ABSTRACT FROM AUTHOR]

This article reviews some results of the SAGEX programme that have developed in the understanding of the interplay of supersymmetry and modular covariance of scattering amplitudes in type IIB superstring theory and its holographic image in N = 4 supersymmetric Yang–Mills theory (SYM). The first section includes the determination of exact expressions for BPS interactions in the low-energy expansion of type IIB superstring amplitudes. The second section concerns properties of a certain class of integrated correlators in N = 4 SYM with arbitrary classical gauge group that are exactly determined by supersymmetric localisation. Not only do these reproduce known features of perturbative and non-perturbative N = 4 SYM for any classical gauge group, but they have large- N expansions that are in accord with expectations based on the holographic correspondence with superstring theory. The final section focusses on modular graph functions. These are modular functions that are closely associated with coefficients in the low-energy expansion of superstring perturbation theory and have recently received quite a lot of interest in both the physics and mathematics literature. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two analogues of the Sturm bound for modular forms in the function field setting. In the case of mixed characteristic, we prove that any harmonic cochain is uniquely determined by an explicit finite number of its first Fourier coefficients where our bound is much smaller than the ones in the literature. A similar bound is derived for generators of the Hecke algebra on harmonic cochains. As an application, we present a computational criterion for checking whether two elliptic curves over the rational function field F q (θ) with same conductor are isogenous. In the case of equal characteristic, we also prove that any Drinfeld modular form is uniquely determined by an explicit finite number of its first coefficients in the t -expansion. [ABSTRACT FROM AUTHOR]

In this paper, we address the case of a particular class of function referred to as the rational equivariant functions. We investigate which elliptic zeta functions arising from integrals of power of ℘, where ℘ is the Weierstrass ℘-function attached to a rank two lattice of C , yield rational equivariant functions. Our concern in this survey is to provide certain examples of rational equivariant functions. In this sense, we establish a criterion in order to determine the rationality of equivariant functions derived from ratios of modular functions of low weight. Modular forms play an important role in number theory and many areas of mathematics and physics. [ABSTRACT FROM AUTHOR]

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p (n) , the number of partitions of n. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. [ABSTRACT FROM AUTHOR]

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest. [ABSTRACT FROM AUTHOR]

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type A 2 and B 2 . This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze Z ^ -invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing H -graphs. Along the way, our method clarifies previous results on depth two quantum modularity. [ABSTRACT FROM AUTHOR]

Let p be a prime ≥ 5 . We establish explicit rates of overconvergence for some members of the "Eisenstein family", notably for the p-adic modular function V (E (1 , 0) ∗) / E (1 , 0) ∗ (V the p-adic Frobenius operator) that plays a pivotal role in Coleman's theory of p-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of p-adic modular functions of form V (E k) / E k where E k is the classical Eisenstein series of level 1 and weight k divisible by p - 1 . Under certain conditions, we extend the latter result to a vast generalization of a theorem of Coleman–Wan regarding the rate of overconvergence of V (E p - 1) / E p - 1 . We also comment on previous results in the literature. These include applications of our results for the primes 5 and 7. [ABSTRACT FROM AUTHOR]

The article discusses the p-adic Maass-Shimura operator and its role in the construction of p-adic L-functions over imaginary quadratic fields K. Other topics include the horizontal lifting of the Hodge-Tate filtration, the equation of the Hodge-Tate sequence, the characteristics of a relative Hodge-Tate decomposition, and the satisfactory properties of the operator like algebraicity.

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular j function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on j with algebraic coefficients. [ABSTRACT FROM AUTHOR]

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels. [ABSTRACT FROM AUTHOR]

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions σ ψ sm (n). More precisely, in terms of the weight 2 quasimodular Eisenstein series E 2 (τ) and a generic Shimura theta function 𝜃 ψ (τ) , we show that there is a constant α ψ for which ε ψ + (τ) : = α ψ ⋅ E 2 (τ) 𝜃 ψ (τ) + 1 𝜃 ψ (τ) ∑ n = 1 ∞ σ ψ sm (n) q n is a half integral weight (polar) mock modular form. These include generating functions for combinatorial objects such as the Andrews s p t -function and the "consecutive parts" partition function. Finally, in analogy with Serre's result that the weight 2 Eisenstein series is a p -adic modular form, we show that these forms possess canonical congruences with modular forms. [ABSTRACT FROM AUTHOR]

One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and sequences. With particular attention to algorithmic aspects, we study various connections between these two worlds. Applications concern partition congruences, Fricke–Klein relations, irrationality proofs a la Beukers, or approximations to pi studied by Ramanujan and the Borweins. As a major ingredient to a "first guess, then prove" strategy, a new algorithm for proving differential equations for modular forms is introduced. [ABSTRACT FROM AUTHOR]

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as q tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function f (q) were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as q tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity. [ABSTRACT FROM AUTHOR]

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 ℤp 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. [ABSTRACT FROM AUTHOR]

We derive a precise asymptotic expansion of the complete Kähler-Einstein metric on the punctured Riemann sphere with three or more omitting points. By using the Schwarzian derivative, we prove that the coefficients of the expansion are polynomials on the two parameters which are uniquely determined by the omitting points. Furthermore, we use the modular form and the Schwarzian derivative to explicitly determine the coefficients in the expansion of the complete Kähler-Einstein metric for the punctured Riemann sphere with 3, 4, 6, or 12 omitting points. [ABSTRACT FROM AUTHOR]

For a positive integer N , let C (N) be the subgroup of J 0 (N) generated by the equivalence classes of cuspidal divisors of degree 0 and C (N) (Q) : = C (N) ∩ J 0 (N) (Q) be its Q -rational subgroup. Let also C Q (N) be the subgroup of C (N) (Q) generated by Q -rational cuspidal divisors. We prove that when N = n 2 M for some integer n dividing 24 and some squarefree integer M , the two groups C (N) (Q) and C Q (N) are equal. To achieve this, we show that all modular units on X 0 (N) on such N are products of functions of the form η (m τ + k / h) , m h 2 | N and k ∈ Z and determine the necessary and sufficient conditions for products of such functions to be modular units on X 0 (N). [ABSTRACT FROM AUTHOR]

Following a suggestion of Peter Scholze, we construct an action of Gm on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p-adic modular curve whose ring of functions is Serre's space of p-adic modular functions. This action is a local, p-adic analog of a global, archimedean action of the circle group S1 on the lattice-unstable locus of the modular curve over C. To construct the Gm-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q; along the way we also prove a natural generalization of Dwork's equation τ = log q for extensions of Qp/Zp by μp∞ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of Gm integrates the differential operator \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p-adic L-functions. [ABSTRACT FROM AUTHOR]

This paper concerns the study of the Schwartz differential equation { h , τ } = s ⁢ E 4 ⁡ (τ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} , where E 4 {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL 2 ⁡ (ℤ) {\operatorname{SL}_{2}({\mathbb{Z}})}. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL 2 ⁡ (ℤ) {\operatorname{SL}_{2}({\mathbb{Z}})}. This also leads to the solutions to the Fuchsian differential equation y ′′ + s ⁢ E 4 ⁡ y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0}. [ABSTRACT FROM AUTHOR]

There are two elementary methods for constructing modular forms that dominate in literature. One of them uses automorphic Poincaré series and the other one theta functions. We start a third elementary approach to modular forms using rational functions that have certain properties regarding pole distribution and growth. We prove modularity with contour integration methods and Weil's converse theorem, without using the classical formalism of Eisenstein series and L-functions. [ABSTRACT FROM AUTHOR]

We give congruences modulo powers of p ∈ { 3 , 5 , 7 } for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and the first author and continuing work done by the authors and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at ∞. [ABSTRACT FROM AUTHOR]

We show that the generating series of traces of reciprocal singular moduli is a mixed mock modular form of weight 3/2 whose shadow is given by a linear combination of products of unary and binary theta functions. To prove these results, we extend the Kudla–Millson theta lift of Bruinier and Funke to meromorphic modular functions. [ABSTRACT FROM AUTHOR]

Quasi-characters are vector-valued modular functions having an integral, but not necessarily positive, q-expansion. Using modular differential equations, a complete classification has been provided in arXiv:1810.09472 for the case of two characters. These in turn generate all possible admissible characters, of arbitrary Wronskian index, in order two. Here we initiate a study of the three-character case. We conjecture several infinite families of quasi-characters and show in examples that their linear combinations can generate admissible characters with arbitrarily large Wronskian index. The structure is completely different from the order two case, and the novel coset construction of arXiv:1602.01022 plays a key role in discovering the appropriate families. Using even unimodular lattices, we construct some explicit three-character CFT corresponding to the new admissible characters. [ABSTRACT FROM AUTHOR]

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi–Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4–D2–D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor D , at the large volume attractor point. For D irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on D and is therefore known to be modular. Instead, when D is the sum of n irreducible divisors D i , we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary n by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth n - 1 . [ABSTRACT FROM AUTHOR]

This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms--Dyson called this 'a challenge for the future' at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later in 2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann-Rolen, Choi-Lim-Rhoades and Griffin-Ono-Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. [ABSTRACT FROM AUTHOR]

In his famous 2007 paper on three-dimensional quantum gravity, Witten defined candidates for the partition functions Z k (q) = ∑ n = - k ∞ w k (n) q n of potential extremal conformal field theories (CFTs) with central charges of the form c = 24 k . Although such CFTs remain elusive, he proved that these modular functions are well defined. In this note, we point out several explicit representations of these functions. These involve the partition function p(n), Faber polynomials, traces of singular moduli, and Rademacher sums. Furthermore, for each prime p ≤ 11 , the p series Z k (q) , where k ∈ { 1 , ⋯ , p - 1 } ∪ { p + 1 } , possess a Ramanujan congruence. More precisely, for every non-zero integer n we have that w k (p n) ≡ 0 (mod 2 11) if p = 2 , (mod 3 5) if p = 3 , (mod 5 2) if p = 5 , (mod p) if p = 7 , 11. [ABSTRACT FROM AUTHOR]

The values of the classical j -invariant at CM points are called singular moduli. Zagier [15] proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier-Funke [1] generalized his result to the sums of the values at Heegner points of modular functions on modular curves of arbitrary genus. Extending the work of [9] , we construct the real-valued class invariants by using the modular functions defined by Weber's resolvents and identify their Galois traces with Fourier coefficients of a weight 3/2 weakly holomorphic modular form by using Shimura's reciprocity law. [ABSTRACT FROM AUTHOR]

We show that if p is a prime, then all elliptic curves defined over the cyclotomic Zp-extension of Q are modular. [ABSTRACT FROM AUTHOR]

Abstract In this work, we study various properties of two families of normal subgroups of the Hecke group Γ 0 (3) and their associated modular curves and function fields. As consequences, we obtain the stabilizer subgroups of the N -th roots of a uniformizer for X (Γ 0 (3)) , and realize such N -th roots as uniformizers for the associated modular curves to their stabilizer groups. [ABSTRACT FROM AUTHOR]

We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose q-expansions satisfy the following: f k (A ; τ) : = q - k (1 + a (1) q + a (2) q 2 + ⋯) + O (q) , where a(n) are numbers satisfying a certain analytic condition. We show that the zeros of such f k (τ) in the fundamental domain of SL 2 (Z) lie on | τ | = 1 and are transcendental. We recover as a special case earlier work of Witten on extremal "partition" functions Z k (τ) . These functions were originally conceived as possible generalizations of constructions in three-dimensional quantum gravity. [ABSTRACT FROM AUTHOR]

We prove divisibility results for the Fourier coefficients of canonical basis elements for the spaces of weakly holomorphic modular forms of weight 0 and levels 6, 10, 12, and 18 with poles only at the cusp at infinity. In addition, we show that these Fourier coefficients satisfy Zagier duality in all weights, and give a general formula for the generating functions of such canonical bases for all genus zero levels. [ABSTRACT FROM AUTHOR]

Some properties of the modular lambda function that are similar to those of the modular invariant functions are proved. An algorithm for constructing the minimal polynomial for the values of the lambda function at the points of imaginary quadratic fields is presented; the numbers conjugate to these values are given. [ABSTRACT FROM AUTHOR]

We study the irreducible algebraic equation x n + a 1 ⁢ x n - 1 + ⋯ + a n = 0 , with n ≥ 4 , x^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0,\quad\text{with ${n\geq 4}$,} on the differential field ( 𝔽 = ℂ ⁢ ( t ) , δ = d d ⁢ t ) {(\mathbb{F}=\mathbb{C}(t),\delta=\frac{d}{dt})}.We assume that a root of the equation is a solution to the Riccati differential equation u ′ + B 0 + B 1 ⁢ u + B 2 ⁢ u 2 = 0 {u^{\prime}+B_{0}+B_{1}u+B_{2}u^{2}=0} ,where B 0 {B_{0}} , B 1 {B_{1}} , B 2 {B_{2}} are in 𝔽 {\mathbb{F}}. We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial F n ⁢ ( x , y ) ∈ ℂ ⁢ ( x ) ⁢ [ y ] {F_{n}(x,y)\in\mathbb{C}(x)[y]} such that for almost T ∈ 𝔽 ∖ ℂ {T\in\mathbb{F}\setminus\mathbb{C}} , the algebraic equation F n ⁢ ( x , T ) = 0 {F_{n}(x,T)=0} is of the same type as the above stated algebraic equation.In other words, all its roots are solutions to the same Riccati equation.On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup Γ ⁢ ( 2 ) {\Gamma(2)} , whose roots satisfy a common Riccati equation on the differential field ( ℂ ⁢ ( E 2 , E 4 , E 6 ) , d d ⁢ τ ) {(\mathbb{C}(E_{2},E_{4},E_{6}),\frac{d}{d\tau})} , with E i ⁢ ( τ ) {E_{i}(\tau)} being the Eisenstein series of weight i.These solutions are related to a Darboux–Halphen system.Finally, we deal with the following problem: For which “potential” q ∈ ℂ ⁢ ( ℘ , ℘ ′ ) {q\in\mathbb{C}(\wp,\wp^{\prime})} does the Riccati equation d ⁢ Y d ⁢ z + Y 2 = q {\frac{dY}{dz}+Y^{2}=q} admit algebraic solutions over the differential field ℂ ⁢ ( ℘ , ℘ ′ ) {\mathbb{C}(\wp,\wp^{\prime})} , with ℘ {\wp} being the classical Weierstrass function?We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial Φ ⁢ ( x ) {\Phi(x)} of an algebraic solution u must have a vanishing fourth transvectant τ 4 ⁢ ( Φ ) ⁢ ( x ) {\tau_{4}(\Phi)(x)}. [ABSTRACT FROM AUTHOR]

Let be the number of -regular partitions of . Recently, Hou et al. established several infinite families of congruences for modulo , where and . In this paper, using the vanishing property given by Hou et al., we prove an infinite family of congruence for modulo . Moreover, for and , we obtain three infinite families of congruences for modulo and respectively using the theory of Hecke eigenforms. [ABSTRACT FROM AUTHOR]

In this paper, we study two families of normal subgroups of Γ 0 ( 2 ) with abelian quotients and their associated modular curves. They are similar to Fermat groups and Fermat curves in some aspects but very different in other aspects. Almost all of them are non-congruence subgroups. These modular curves are either projective lines or hyperelliptic curves. There are few modular forms of weight 1 for these groups. We also determine their cuspidal divisor class groups and show that these groups are finite. [ABSTRACT FROM AUTHOR]

We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects. [ABSTRACT FROM AUTHOR]

Let M be a given positive integer and r = ( r δ ) δ | M a sequence indexed by the positive divisors δ of M . In this paper we present an algorithm that takes as input a generating function of the form ∑ n = 0 ∞ a r ( n ) q n : = ∏ δ | M ∏ n = 1 ∞ ( 1 − q δ n ) r δ and positive integers m , N and t ∈ { 0 , … , m − 1 } . Given this data we compute a set P m , r ( t ) which contains t and is uniquely defined by m , r and t . Next we decide if there exists a sequence ( s δ ) δ | N indexed by the positive divisors δ of N , and modular functions b 1 , … , b k on Γ 0 ( N ) (where each b j equals the product of finitely many terms from { q δ / 24 ∏ n = 1 ∞ ( 1 − q δ n ) : δ | N } ), such that: q α ∏ δ | N ∏ n = 1 ∞ ( 1 − q δ n ) s δ × ∏ t ′ ∈ P m , r ( t ) ∑ n = 0 ∞ a ( m n + t ′ ) q n = c 1 b 1 + ⋯ + c k b k for some c 1 , ⋯ , c k ∈ Q and α : = ∑ δ | N δ s δ 24 + ∑ t ′ ∈ P m , r ( t ) 24 t ′ + ∑ δ | M δ r δ 24 m . Our algorithm builds on work by Rademacher (1942) , Newman (1959) , and Kolberg (1957) . [ABSTRACT FROM AUTHOR]