Theta Functions Of Order Research Articles

On Frobenius' Theta Formula

Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which follows from Riemann's theta formula. In a 2004 paper Grushevsky gives a different characterization in terms of cubic equations in second order theta functions. In this note we first show the connection between the methods by proving that Grushevsky's cubic equations are strictly related to Frobenius' theta formula and we then give a new proof of Mumford's characterization via Gunning's multisecant formula.

New Fifth and Seventh Order Mock Theta Function Identities

We give simple proofs of Hecke-Rogers indefinite binary theta series identities for the two Ramanujan fifth order mock theta functions $\chi_0(q)$ and $\chi_1(q)$ and all three of Ramanujan's seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related.

A new approach in interpreting some mock theta functions

We are applying a new approach to interpret the generalized version of order 8 mock theta functions [Formula: see text] and [Formula: see text] in terms of [Formula: see text]-color partitions. In literature, we find the interpretations of these functions in terms of split [Formula: see text]-color partitions and signed partitions. In addition, we are interpreting the generalized versions of two mock theta functions of order 6 and two of order 5 in terms of [Formula: see text]-color partitions. A mock theta function of order 3 is interpreted in terms of [Formula: see text]-color partitions. We further extend our results using lattice paths.

Quantum modularity of mock theta functions of order 2

In [9], we computed shadows of the second order mock theta functions and showed that they are essentially same with the shadow of a mock theta function related to the Mathieu moonshine phenomenon. In this paper, we further survey the second order mock theta functions on their quantum modularity and their behavior in the lower half plane.

A Note on Certain Properties of Mock Theta Functions of Order Eight

In this paper, we have developed a non-homogeneous q- difierence equa- tion of flrst order for the generalized Mock theta function of order eight and besides these established limiting case of Mock theta functions of order eight. We have also established identities for Partial Mock theta function and Mock theta function of order eight and provided a number of cases of the identities.

A report on complete mock theta functions of order eight

The aim of the present paper is to establish new representation of complete mock theta functions of order eight, certain relations between complete mock theta functions of order eight and some relations between complete mock theta functions and mock theta functions of order eight

TWO CONGRUENCES FOR APPELL–LERCH SUMS

Two congruences are proved for an infinite family of Appell–Lerch sums. As corollaries, special cases imply congruences for some of the mock theta functions of order two, six and eight.

CERTAIN RELATIONS FOR MOCK THETA FUNCTIONS OF ORDER EIGHT

The aim of the present paper is to establish certain relations for partial mock theta functions and mock theta functions of order eight with other partial mock theta functions and mock theta functions of order two, six, eight and ten respectively.

On theta functions of order 4

We prove that the fourth powers of theta functions with even characteristics form a basis of the space of even theta functions of order four on a principally polarized Abelian variety without vanishing theta-null.

Nonabelian theta functions of positive genus

'. Let Cg be a smooth projective irreducible curve over C of genus g ≥ 1 and let {p 1 ,..., p s } be a set of distinct points on Cg. We fix a nonnegative integer l and denote by M g (p, λ) the moduli space of parabolic semistable vector bundles of rank r on Cg with trivial determinant and fixed parabolic structure of type A = (λ 1 ,..., λ s ) at p = (p 1 ,..., p s ), where each weight λ i is in P l (SL(r)). On M g (p, λ) there is a canonical line bundle L(λ, l), whose global sections are called generalized parabolic SL(r)-theta functions of order l. In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.

PARTIAL SECOND ORDER MOCK THETA FUNCTIONS, THEIR EXPANSIONS AND PADE APPROXIMANTS

By proving a summation formula, we enumerate the expansions for the mock theta functions of order 2 in terms of partial mock theta functions of order 2, 3 and 6. We show a relation between Ramanujan`s -function and his sixth order mock theta functions. In addition, we also give the continued fraction representation for and 2nd order mock theta functions and approximants.

Ramanujan Modular Forms and the Klein Quartic

In one of his notebooks, Ramanujan gave some algebraic relations between three theta functions of order 7. We describe the au- tomorphic character of a vector-valued mapping constructed from these theta series. This provides a systematic way to establish old and new identities on modular forms for the congruence subgroup of level 7, above all, a parametrization of the Klein quartic. From a historical point of view, this shows that Ramanujan discovered the main properties of this curve with his own means. As an application, we introduce an L-series in four dierent ways, generating the number of points of the Klein quartic over finite fields. From this, we derive the structure of the Jacobian of a suitable form of the Klein quartic over finite fields and some congruence properties on the number of its points.

Certain continued fraction representaions for functions associated with mock theta functions of order three

Certain continued fraction representaions for functions associated with mock theta functions of order three

Some Properties of Second Order Theta Functions o Prym Varieties

Let P ∪ P′ be the two component Prym variety associated to an etale double cover C C of a non-hyperelliptic curve of genus g ≥6 and let |2Ξ0| and |2Ξ′0| be the linear systems of second order theta divisors on P and P′ respectively. The component P′ contains canonically the Prym curve C. We show that the base locus of the subseries of divisors containing C ⊂ P′ is exactlythe curve C. We also prove canonical isomorphisms between some subseries of |2Ξ0| and |2Ξ′0| and some subseries of second order theta divisors on the Jacobian of C.

Identities Involving Partial Mock Theta Functions of Order Five

In this paper we have given transformations for the partial mock theta functions of order five and also some identities between these partial mock theta functions analogous to the identities given by Ramanujan.

Separation of variables for quantum integrable systems on elliptic curves

We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group $E_{\tau,\eta}(sl_2)$ by difference operators. One model of statistical mechanics to which this method applies is the IRF model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher order theta functions.

On partial sums of mock theta functions of order three

The object of this paper is to define and study the properties of partial mock theta functions of order three, on the lines Ramanujan had studied partial θ-functions. These new partial functions have been expressed in terms of basic hypergeometric function2Φ1. Their continued fractions representations have also been given.

On certain Ramanujan's mock theta functions

Ramanujan's last gift to the mathematicians was his ingeneous discovery of the mock theta functions of order three, five and seven. Recently, Andrews and Hickerson found a set of seven more functions in Ramanujan's Lost Note Book and formally labelled them as mock theta functions of order six. In this paper the complete forms of these functions have been studied and connected with the bilateral basic hypergeometric series2Ψ2. Several other interesting properties and transformations have also been studied.

Irreducible Representations of the 4-Dimensional Sklyanin Algebra at Points of Infinite Order

In 1982 Sklyanin ( Funct. Anal. Appl. 16 (1982), 27-34) defined a family of graded algebras A( E, τ), depending on an elliptic curve E and a point τ ∈ E which is not 4-torsion. Basic properties of these algebras were established in Smith and Stafford ( Compositio Math. 83 (1992), 259-289) and a study of their representation theory was begun in Levasseur and Smith ( Bull. Soc. Math. France 121 (1993), 35-90). The present paper classifies the finite dimensional simple A-modules when τ is a point of infinite order. Sklyanin ( Funct. Anal. Appl. 17 (1983), 273-284) defines for each k ∈ N a representation of A in a certain k-dimensional subspace of theta functions of order 2( k − 1). We prove that these are irreducible representations, and that any other simple module is obtained by twisting one of these by an automorphism of A. The automorphism group of A is explicitly computed. The method of proof relies on results in Levasseur and Smith. In particular, it is proved that every finite dimensional simple module is a quotient of a line module. An important part of the analysis is a determination of the 1-critical A-modules, and the fact that such a module is (equivalent to) a quotient of a line module by a shifted line module.

Second Order Theta Functions and Vector Bundles Over Jacobi Varieties

Second Order Theta Functions and Vector Bundles Over Jacobi Varieties