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We count the number of critical points of a modular form with real Fourier coefficients in a γ -translate of the standard fundamental domain  ℱ (with γ ∈ SL 2 (ℤ)). Whereas by the valence formula the (weighted) number of zeros of this modular form in  γ ℱ is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of  ℱ , the value of  γ − 1 (∞) and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form. [ABSTRACT FROM AUTHOR]

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n > 2 is prime if and only if... [ABSTRACT FROM AUTHOR]

Here we investigate the q -series U a (q) = ∑ n = 0 ∞ M O (a ; n) q n : = ∑ 0 < k 1 < k 2 < ⋯ < k a q k 1 + k 2 + ⋯ + k a (1 − q k 1 ) 2 (1 − q k 2 ) 2 ⋯ (1 − q k a ) 2 , U a ⋆ (q) = ∑ n = 0 ∞ M (a ; n) q n : = ∑ 1 ≤ k 1 ≤ k 2 ≤ ⋯ ≤ k a q k 1 + k 2 + ⋯ + k a (1 − q k 1 ) 2 (1 − q k 2 ) 2 ⋯ (1 − q k a ) 2 . MacMahon introduced the U a (q) in his seminal work on partitions and divisor functions. Recent works show that these series are sums of quasimodular forms with weights ≤ 2 a. We make this explicit by describing them in terms of Eisenstein series. We use these formulas to obtain explicit and general congruences for the coefficients M O (a ; n) and M (a ; n). Notably, we prove the conjecture of Amdeberhan-Andrews-Tauraso as the m = 0 special case of the infinite family of congruences M O (11 m + 10 ; 11 n + 7) ≡ 0 (mod 11) , and we prove that M O (17 m + 16 ; 17 n + 15) ≡ 0 (mod 17). We obtain further formulae using the limiting behavior of these series. For n ≤ a + ( a + 1 2 ) , we obtain a "hook length" formula for M O (a ; n) , and for n ≤ 2 a , we find that M (a ; n) = ( a + n − 1 n − a ) + ( a + n − 2 n − a − 1 ). [ABSTRACT FROM AUTHOR]

Simple Hurwitz numbers are classical invariants in enumerative geometry counting branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several modifications of this notion for genus 0 base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the Harish-Chandra-Itzykson-Zuber integral in random matrix theory and strictlymonotone Hurwitz numbers which enumerate certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers for arbitrary genera and any number of arbitrary but fixed ramification profiles. This yields generalisations of several results known for Hurwitz numbers. When the target surface is of genus one, we show that the generating series of these interpolated Hurwitz numbers are quasimodular forms. In the case that all ramification is simple, we refine this result by writing this series as a sum of quasimodular forms corresponding to tropical covers weighted by Gromov-Witten invariants. Moreover, we derive a quantum curve for monotone and Grothendieck dessins d'enfants Hurwitz numbers for arbitrary genera and one arbitrary but fixed ramification profile. Thus, we obtain spectral curves via the semi-classical limit as input data for the Chekhov-Eynard-Orantin (CEO) topological recursion. Astonishingly, we find that the CEO topological recursion for the genus 1 spectral curve of the strictly monotone Hurwitz numbers computes the monotone Hurwitz numbers in genus 0. Thus, we give a new proof that monotone Hurwitz numbers satisfy CEO topological recursion. This points to an unknown relation between those enumerative invariants. Finally, specializing to target surface P¹, we find recursions for monotone and Grothendieck dessins d'enfants double Hurwitz numbers, which enables the computation of the respective Hurwitz numbers for any genera with one arbitrary but fixed ramification profile. [ABSTRACT FROM AUTHOR]

In this paper, we consider twisted Dirichlet series attached to quasimodular forms, study their analytic properties, and prove an analogue of Weil's converse theorem for quasimodular forms over congruence subgroups. We also give some applications of our results to a certain q-series and sign changes of the Fourier coefficients of quasimodular forms. [ABSTRACT FROM AUTHOR]

We study generating series of Gromov-Witten invariants of E P¹ and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a pearl chain. The individual summands are - just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals - complex analytic path integrals involving a product of propagators (equal to the Weierstrass-)-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in ET P1 T of so-called leaky degree. [ABSTRACT FROM AUTHOR]

Jacobi-like forms generalize Jacobi forms, and heat operators on Jacobi forms can be extended to heat operators on Jacobi-like forms. Quasimodular forms correspond to quasimodular polynomials and modular polynomials, and they can be lifted to Jacobi-like forms. We obtain an explicit formula for a differential operator on the space of quasimodular polynomials which is compatible with a heat operators on the space of Jacobi-like forms. We also determine a linear map on the space of modular polynomials that is compatible with this differential operator. [ABSTRACT FROM AUTHOR]

We study a family of residual crank generating functions defined on overpartitions, the so-called k th residual cranks. Specifically, the moment generating functions associated to these cranks exhibit quasimodularity properties which are dependent on the choice of k. We also show that the second moments of these cranks admit a combinatoric interpretation as weighted overpartition counts. This interpretation gives a refinement to an existing inequality between crank moments of differing modulus k. [ABSTRACT FROM AUTHOR]

Recently, many identities for the convolution sum W a , b (n) : = ∑ a ℓ + b m = n σ (ℓ) σ (m) of the divisor function σ (n) : = ∑ d | n d have been obtained since Royer obtained by the theory of quasimodular forms. We also present new identities for a b = 1 7 , 2 9 , 4 1 , 4 7 , 5 9 and 7 1 by using quasimodular forms. [ABSTRACT FROM AUTHOR]

We prove the quasimodularity of generating functions for counting torus covers, with and without Siegel-Veech weight. Our proof is based on analyzing decompositions of flat surfaces into horizontal cylinders. The quasimodularity arises from representing the generating series as a contour integral of quasi-elliptic functions. This provides an alternative proof of the quasimodularity results of Bloch-Okounkov, Eskin-Okounkov and Chen-Möller-Zagier, and generalizes the results of Böhm-Bringmann-Buchholz-Markwig for simple ramification covers. [ABSTRACT FROM AUTHOR]

We consider Dirichlet series attached to a quasimodular form, study its analytic properties, and generalize Hecke's converse theorem to quasimodular forms of any weight and depth over SL 2 (Z). Then we discuss some applications of our results to a certain q -series and to sign changes of the Fourier coefficients of quasimodular forms. [ABSTRACT FROM AUTHOR]

Jacobi-like forms for a discrete subgroup Γ of SL(2,R) are formal power series which generalize Jacobi forms, and they are in one-to-one correspondence with automorphic pseudodifferential operators for Γ. The well-known Cohen-Kuznetsov lifting of a modular form f provides a Jacobi-like form and therefore an automorphic pseudodifferential operator associated to f. Given a pair (λ,μ) of integers, automorphic pseudodifferential operators can be extended to those of mixed weight. We show that each coefficient of an automorphic pseudodifferential operator of mixed weight is a quasimodular form and prove the existence of a lifting of Cohen-Kuznetsov type for each quasimodular form. [ABSTRACT FROM AUTHOR]

We establish a correspondence between vector-valued modular forms with respect to a symmetric tensor representation and quasimodular forms. This is carried out by first obtaining an explicit isomorphism between the space of vector-valued modular forms with respect to a symmetric tensor representation and the space of finite sequences of modular forms of certain type. This isomorphism uses Rankin–Cohen brackets and extends a result of Kuga and Shimura, who considered the case of vector-valued modular forms of weight two. We also obtain a correspondence between such vector-valued modular forms and weak Jacobi forms. [ABSTRACT FROM AUTHOR]

Given positive integersand, a vectorofand a symmetric positive definite integralmatrixwhose diagonal entries are even, we consider an associated series similar to theta series and prove that this series is a modular form of weightfor a congruence subgroup with character. [ABSTRACT FROM PUBLISHER]

A quasimodular form $$\phi $$ of depth at most $$m$$ corresponds to holomorphic functions $$\phi _0, \phi _1, \ldots , \phi _m$$ . Given nonnegative integers $$\alpha $$ and $$\nu $$ with $$\nu \le m$$ , we introduce a linear differential operator $$\mathcal D_{\phi }^{\alpha , \nu }$$ of order $$\nu $$ on modular forms whose coefficients are given in terms of derivatives of the functions $$\phi _k$$ . We then show that Rankin-Cohen brackets of modular forms can be expressed in terms of such operators. As an application, we obtain differential operators associated to certain theta series studied by Dong and Mason. [ABSTRACT FROM AUTHOR]

We study various properties of quasimodular forms by using their connections with Jacobi-like forms. Such connections are made by identifying quasimodular forms for a discrete subgroup $$\Gamma $$ of $$SL(2, {\mathbb {R}})$$ with certain polynomials over the ring of holomorphic functions of the Poincaré upper half plane that are $$\Gamma $$ -invariant. We consider a surjective map from Jacobi-like forms to quasimodular forms and prove that it has a right inverse, which may be regarded as a lifting from quasimodular forms to Jacobi-like forms. [ABSTRACT FROM AUTHOR]

We derive a multiplicity one result for quasimodular eigenforms on SL2(ℤ) by completing the description of all quasimodular eigenforms. As applications of our results, we complete the answer to the question of finding all quasimodular eigenforms arising as a product of two eigenforms and also give an estimate for the size of the maximum gap in the sequence of the Fourier coefficients of a quasimodular form. [ABSTRACT FROM AUTHOR]

Using quasimodular forms with respect to Γ0(4) we find exact relations between the M2-rank for partitions without repeated odd parts and three residual cranks. From these identities we are able to deduce various congruences mod 3 and 5 between the rank and crank moments. In turn, these congruences give congruences for (n), the number of occurrences of smallest parts in the partitions of n with smallest part even and without repeated odd parts, and for the higher-order analog (n). [ABSTRACT FROM AUTHOR]

In this paper, we establish automorphic properties and asymptotic behaviors of characters due to Kac-Wakimoto pertaining to $$s\ell (m|n)^\wedge $$ highest weight modules in the case $$m=n$$ , extending work of Bringmann and Ono (Math Ann 345:547-558, ) and Bringmann and Folsom (submitted) which pertains to the case $$m>n$$ . [ABSTRACT FROM AUTHOR]

We evaluate the convolution sums Σl+27m=n σ(l)σ(m), Σ5l+8m=n σ(l)σ(m), Σ5l+11m=n σ(l)σ(m), Σ5l+m=n σ(l)τ2,11 (m), Σ11l+5m=n σ(l)τ2,11 (m), Σ55l+m=n σ(l)τ2,11 (m), Σl+5m=n τ2,11(l)τ2,11 (m), Σ3l+9m=n σ(l)σ(m), Σ4l+10m=n σ(l)σ(m), Σl+5m=n σ(l)τ2,11 (m), Σl+40m=n σ(l)σ(m), Σl+55m=n σ(l)σ(m), and for all positive integers n using the theory of quasimodular forms, we determine the number of representations of a positive integer n by the forms x12 + x1 x2 + x22 + x32 + x3 x4 + x42 + 9 (x52 + x52 x6 + x62 + x72 + x7 x8 + x82), x12 + 2x22 + x32 + 2x42 + 5 (x52 + 2x62 + x72 + 2x82), x12 + x1 x2 + 3x22 + 3x22 + x3 x4 + 3x22 + 5 (x52 + x5 x6 + 3x62 + x72 + x7 x8 + 3x82). [ABSTRACT FROM AUTHOR]

The theory of quasimodular forms is used to evaluate the convolution sums ... σ(l) σ(m), ... σ(l) σ(m) and ... σ(l) σ(m) for all positive integers n. As a consequence, the number of representations of a positive integer n by the octonary quadratic form x12 + x22 + x32 + x42 + 5(x52 + x62 + x72 + x82 is determined. [ABSTRACT FROM AUTHOR]

We evaluate the convolution sums ∑l,m∈ℕ,l+15m=nσ(l)σ(m) and ∑l,m∈ℕ,3l+5m=nσ(l)σ(m) for all n ∈ ℕ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer n by the form We also determine the number of representations of positive integers by the quadratic form by using the convolution sums obtained earlier by Alaca, Alaca and Williams [Evaluation of the convolution sums ∑l+6m=nσ(l)σ(m) and ∑2l+3m=nσ(l)σ(m), J. Number Theory 124(2) (2007) 491-510; Evaluation of the convolution sums ∑l+24m=nσ(l)σ(m) and ∑3l+8m=nσ(l)σ(m), Math. J. Okayama Univ. 49 (2007) 93-111]. [ABSTRACT FROM AUTHOR]

Modular polynomials are polynomials whose coefficients are modular forms of certain type, and they are in one-to-one correspondence with quasimodular polynomials that are naturally associated to quasimodular forms. In addition to the formal derivative operator ∂X, there is a differential operator 𝒟λon quasimodular polynomials corresponding to the derivative operator on quasimodular forms. We construct linear maps of quasimodular polynomials corresponding to certain linear maps of modular polynomials and prove that such maps reduce to the operators 𝒟λand ∂Xin special cases. [ABSTRACT FROM PUBLISHER]

Families of quasimodular forms arise naturally in many situations, such as curve counting on Abelian surfaces and counting ramified covers of orbifolds. In many cases the family of quasimodular forms naturally arises as the coefficients of a Taylor expansion of a Jacobi form. In this paper we give examples of such expansions that arise in the study of partition statistics. The crank partition statistic has gathered much interest recently. For instance, Atkin and Garvan showed that the generating functions for the moments of the crank statistic are quasimodular forms. The two-variable generating function for the crank partition statistic is a Jacobi form. Exploiting the structure inherent in the Jacobi theta function, we construct explicit expressions for the functions of Atkin and Garvan. Furthermore, this perspective opens the door for further investigation, including a study of the moments in arithmetic progressions. We conduct a thorough study of the crank statistic restricted to a residue class modulo 2. [ABSTRACT FROM AUTHOR]

We investigate the cases for which products of two quasimodular or nearly holomor-phic eigenforms are eigenforms. We also generalize the results of Ghate [On products of eigenforms, Acta Arith. 102 (2002) 27-44] to the case of Rankin-Cohen brackets. [ABSTRACT FROM AUTHOR]

We construct linear maps from the spaces of quasimodular forms for a discrete subgroup Γ of SL(2,ℝ) to some cohomology spaces of the group Γ and prove that these maps are equivariant with respect to appropriate Hecke operator actions. The results are obtained by using the fact that there is a correspondence between quasimodular forms and certain finite sequences of modular forms. [ABSTRACT FROM AUTHOR]

Abstract: Andrewsʼ spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only known previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived. [Copyright &y& Elsevier]

Abstract: Given modular forms f and g of weights k and ℓ, respectively, their Rankin–Cohen bracket corresponding to a nonnegative integer n is a modular form of weight , and it is given as a linear combination of the products of the form for . We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin–Cohen brackets. [ABSTRACT FROM AUTHOR]

We introduce a differential operator on quasimodular polynomials that corresponds to the derivative operator on quasimodular forms. We then prove that such a differential operator is compatible with a heat operator on Jacobi-like forms in certain cases. These results show in those cases that the derivative operator on quasimodular forms corresponds to a heat operator on Jacobi-like forms. [ABSTRACT FROM AUTHOR]

Modular forms for a discrete subgroup Γ of SL.2; R/ can be identified with holomorphic sections of line bundles over the modular curve U corresponding to Γ, and quasimodular forms generalize modular forms. We construct vector bundles over U whose sections can be identified with quasimodular forms for Γ. [ABSTRACT FROM AUTHOR]

Abstract: We describe the additive structure of the graded ring of quasimodular forms over any discrete and cocompact group We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight k. This number is constant for k sufficiently large and equals where I and are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that is contained in some finitely generated ring of meromorphic quasimodular forms with i.e., the same order of growth as [Copyright &y& Elsevier]

We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular forms. This extension of modular forms has been constructed by Kaneko and Zagier. [ABSTRACT FROM AUTHOR]