Narrow Results

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring of global holomorphic differential operators on J. We construct a free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.

By using the Milnor ring structure of the cohomology of Grassmannian, we prove a vanishing theorem of Witten genus for generalized string complete intersections in products of Grassmannian.

In [Q. Ren, S. Sam, G. Schrader and B. Sturmfels, The universal Kummer threefold, Experiment Math.22(3) (2013) 327–362], the authors conjectured equations for the universal Kummer variety in genus 3 case. Although, most of these equations are obtained from the Fourier–Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic, cf. [A. Coble, Algebraic Geometry and Theta Functions, American Mathematical Society Colloquium Publications, Vol. 10 (American Mathematical Society, 1929)] was obtained differently, cf. [S. Grushevsky and R. Salvati Manni, On Coble's quartic, preprint (2012), arXiv:1212.1895] too. The aim of this paper is to show that Coble's quartic can be obtained as Fourier–Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in [Experiment Math.22(3) (2013) 327–362].

Let $X$ be an abelian surface, and $D$ be the sum of $N(\ge 2)$ distinct theta divisors having normal crossings. We set $M=X-D$. We study the structure of the nonvanishing twisted cohomology group $H^2(M,{ℒ})$, where ${ℒ}$ denotes a locally constant sheaf over $M$ defined by a multiplicative meromorphic function on $M$ infinitely ramified just along the divisor $D$ (as will be seen below, we will take as such a function a product of complex powers of theta functions). The de Rham complex on $X$ with logarithmic poles along $D$, associated to the twisted cohomology groups $H^p(M,{ℒ})$, is $P$-valued, where $P$ denotes a topologically trivial (i.e. Chern class zero) line bundle over $X$ determined by the locally constant sheaf ${ℒ}$. Therefore the main results of this paper, which give us information on the order of poles of meromorphic 2-forms on $X$ generating the cohomology group $H^2(M,{ℒ})$, are divided into Theorems 4.5 and 4.6, according as the de Rham complex on $X$ with logarithmic poles along $D$ takes the values in a holomorphically nontrivial line bundle $P\ne 1$ or a holomorphically trivial one $P=1$ ($1$ denoting the holomorphically trivial line bundle $C\times X$). Such a phenomenon does not occur in the case of the twisted cohomology of complex projective space with hyperplane arrangement.

The Riemann–Wirtinger integral is an analogue of the hypergeometric integral, which is defined on a one-dimensional complex torus. We study the intersection forms on the twisted homology and cohomology groups associated with the Riemann–Wirtinger integral. We derive explicit formulas of some intersection numbers, and apply them to studies of the monodromy representation, connection problems, and contiguity relations.

Let r_k(n) and t_k(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result r_k(8n + k) = c_kt_k(n), which holds for 1 ≤ k ≤ 7, where c_k is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial.

The quintuple product identity was first discovered about 90 years ago. It has been published in many different forms, and at least 29 proofs have been given. We shall give a comprehensive survey of the work on the quintuple product identity, and a detailed analysis of the many proofs.

We give a new proof of Milne's formulas for the number of representations of an integer as a sum of 4m² and 4m(m + 1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne's formulas to Schur Q-polynomials and to correlation functions for continuous dual Hahn polynomials. We also state a new formula for 2m² squares.

The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

Two pairs of inverse relations for elliptic theta functions are established with the method of Fourier series expansion, which allow us to recover many classical results in theta functions. Many nontrivial new theta function identities are discovered. Some curious trigonometric identities are derived.

We find several new parity results for broken 5-diamond, 7-diamond and 11-diamond partitions.

We find several new parity results for 7- and 23-regular partitions.

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. He also proved that . Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for , and . In this paper, we find new congruences for modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for , congruences modulo 2 and 3 for , , and congruences modulo 2 for and . We use simple p-dissections of Ramanujan's theta functions.

Let $\mathbb{Z}$ and $\mathbb{N}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{N}$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $a{x}^2+b{y}^2+c{z}^2+d{w}^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2+dw(w-1)/2$$(x,y,z,w\in \mathbb{Z}$). In this paper, we reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in \mathbb{N}$ and $2\nmid a$. We show that

Andrews defined the combinatorial objects called singular overpartitions denoted by ${\overline{C}}_{k,i}(n)$, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i(modk)$ may be overlined. In this paper, we investigate the arithmetic properties of Andrews singular overpartition pairs. Let ${\overline{A}}_{i,j}^{\delta }(n)$ be the number of overpartition pairs of $n$ in which no part is divisible by $\delta$ and only parts $\equiv \pm i,\pm j(mod\delta )$ may be overlined. We will prove a number of Ramanujan like congruences and infinite families of congruences for ${\overline{A}}_{1,2}^6(n)$ modulo $3,9,18$ and $36$, infinite families of congruences for ${\overline{A}}_{2,4}^8(n)$ modulo $4$ and $8$, infinite families of congruences for ${\overline{A}}_{1,5}^{12}(n)$ modulo $6$ and $9$.

This paper describes the derivation of the level 5 versions of Ramanujan’s system of ordinary differential equations satisfied by the Eisenstein series, $E_2(q),E_4(q)$, and $E_6(q).$

Let $\mathbb{Z}$ and ${\mathbb{Z}}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,n\in {\mathbb{Z}}^{+}$ let $N(a,b,c;n)$ be the number of representations of $n$ by $a{x}^2+b{y}^2+c{z}^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2$$(x,y,z\in \mathbb{Z})$. In this paper, by using Ramanujan’s theta functions $\phi (q)$ and $\psi (q)$ we reveal some general relations between $t(a,b,c;n)$ and $N(a,b,c;8n+a+b+c)$.

Let $\mathbb{Z}$ and $\mathbb{N}$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\dots ,a_k,n\in \mathbb{N}$, let $N(a_1,a_2,\dots ,a_k;n)$ be the number of representations of $n$ by $a_1{x}_1^2+a_2{x}_2^2+\cdots +a_k{x}_k^2$, and let $t(a_1,a_2,\dots ,a_k;n)$ be the number of representations of $n$ by $a_1\frac{x_1(x_1-1)}{2}+a_2\frac{x_2(x_2-1)}{2}+\cdots +a_k\frac{x_k(x_k-1)}{2}$$(x_1,\dots ,x_k\in \mathbb{Z}$). In this paper, by using Ramanujan’s theta functions $\phi (q)$ and $\psi (q)$, we reveal many relations between $t(a_1,a_2,\dots ,a_k;n)$ and $N(a_1,a_2,\dots ,a_k;8n+a_1+\cdots +a_k)$ for $k=3,4$.

Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map ${{𝒮}}_1$) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where Jacobi theta functions are replaced by the theta functions associated to Dirichlet character of prime power moduli, and the level one Hecke eigenforms are replaced by newforms of arbitrary level. Hansen and Naqvi generalized Cipra’s work (on the image of a class of modular forms under the first Shimura map ${{𝒮}}_1$) to cover theta functions associated to Dirichlet characters of arbitrary moduli. In this paper, we show that the earlier results can be modified to get similar results for the $t$th Shimura lifts ${{𝒮}}_t$, for any positive square-free integer $t$.

In the eleventh paper in the series on MacMahon’s partition analysis, Andrews and Paule introduced the $k$-elongated partition diamonds. Recently, they revisited the topic. Let $d_k(n)$ count the partitions obtained by adding the links of the $k$-elongated plane partition diamonds of length $n$. Andrews and Paule obtained several generating functions and congruences for $d_1(n)$, $d_2(n)$, and $d_3(n)$. Da Silva et al. further found many congruences modulo 4, 5, 7, 8, 9, and 11 for $d_k(n)$. In this paper, we extend some individual congruences found by Andrews and Paule and da Silva et al. to their respective families as well as find new families of congruences for $d_k(n)$. We also present a refinement in an existence result for congruences of $d_k(n)$ found by da Silva et al. and prove some new individual as well as a few families of congruences modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64, and 128.