General Abelian Varieties and Theta Function

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan’s identities, Winquist’s identity and many other interesting identities.

The Borwein brothers have introduced and studied three cubic theta functions. Many generalizations of these functions have been studied as well. In this paper, we introduce a new generalization of these functions and establish general formulas that are connecting our functions and Ramanujan's general theta function. Many identities found in the literature follow as a special case of our identities. We further derive general formulas for certain products of theta functions.

Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of(2+1)-dimensional nonlinear partial differential equation. The resulting theory is applied to the(2+1)-dimensional Sawada-Kotera equation, thereby yielding its double periodic wave solutions. The relations between the periodic wave solutions and soliton solutions are rigorously established by a limiting procedure.

In this paper, we use the theta function method [10] to give explicit Doi-Naganuma type maps associated to an imaginary quadratic field K, lifting cusp forms on any congruence subgroup of SL(2, Z) to forms on SL(2, C) automorphic with respect to an appropriate arithmetic discrete subgroup. The case of class number one, and form modular with respect to group Γ0(D) and character χ0 = (–D/*), where –D is the discriminant of K has been treated by Asai [1]. In order to complete his discussion, we must first introduce a more general theta function associated to an indefinite quadratic form (here of type (3, 1)), which we regard as a specialization of a symplectic theta function (see also [4]).

Theta functions have historically played a prominent role in number theory. One such role is the construction of modular forms. In this work, a generalized theta function is used to construct an infinite family of summation identities. Our results grew out of some observations noted during a presentation given by the author at the 1992 AMS-MAA-SIAM Joint Meetings in Baltimore.

One of the most elegant classical results about nonsingular algebraic curves is Riemann’s singularity theorem, which provides a beautiful interpretation of the singularities of Riemann’s theta function in terms of the geometry of special divisors. In recent times, soliton theory has suggested generalizations of the classical geometry. Integrable partial differential equations posed with periodic or scattering boundary conditions lead to Riemann surfaces of infinite, or even “continuously infinite”, genus, and to correspondingly huge Jacobians ([McKT], [EMcK]). More special boundary conditions have been studied via the theory of infinite-dimensional Lie groups and their homogeneous spaces. In this latter approach, one finds a large family of (possibly infinite-dimensional) algebraic varieties that are natural generalizations of the classical Jacobians. The Lie-theoretic aspects of these varieties are, of course, prominent in such work ([SW], [Sch]). In another direction, soliton problems have led naturally to singular limits of smooth Riemann surfaces. While, in the abstract, singular curves and their Jacobians are a subject of current research, soliton theory provides a concrete context in which to study such algebraic varieties.

We study the large N and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for \U0001d4a9 = 4 U(N) super Yang-Mills theory. By means of the factorization property, the large N correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan’s general theta functions and the q-analogues of multiple zeta values (q-MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large N correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large N correlators.

In this paper we present a calculation to study a relationship between the generalized theta function and the Deuring polynomial.

Introduction Riemann surfaces Riemann-Roch theorem Abel-Jacobi theorem and period relations Divisors and theta functions Dimension of the space of theta functions Projective embedding and theta functions (chapter 6) Elliptic curves as the intersection of two quadrics The Fermat curve Discrete subrgroups of SL(sub 2)(R) Riemann surface structure of $\Gamma \setminus \mathcal H^*$ The modular curve X(N) Generalities on abelian varieties The conjecture of Tate Finiteness of isogeny classes (chapter 14) Mordell's conjecture Bibliography Index.

If M is a marked compact Riemann surface of genus g>0 and ξ is a factor of automorphy with characteristic class c(ξ)=r then by the Riemann-Roch theorem γ(ξ)=γ(κξ-1)+r+1-g, where κ is the canonical factor of automorphy; and if r≥2g-1 then c(κξ)=2g-2-r<0 so that γ(κξ-1)=0 and γ(ξ)=r+1-g. Thus all factors of automorphy with characteristic class r admit equally many complex analytic relatively automorphic functions whenever r≥2g-1. Now the set of factors of automorphy with characteristic class r can be parametrized by the complex manifold ℂ g , by associating to any point t∊- g the factor of automorphy ρtζ r where ρ: ℂ g →Hom(Г, ℂ*) is the canonical homomorphism considered previously and ζ is a factor of automorphy representing the point bundle associated to the divisor 1·p0 for the base point p0 of the marked surface M; and it can be asked whether the set of all complex analytic relatively automorphic functions for these factors of automorphy can be parametrized accordingly by the complex manifold ℂg× ℂr+1-g if r≥2g-1. The answer is provided by the following result.

Let M(r) be the moduli space of rank r vector bundles with trivial determinant on a Riemann surface X . This space carries a natural line bundle, the determinant line bundle L . We describe a canonical isomorphism of the space of global sections of L^k with a space known in conformal field theory as the ``space of conformal blocks, which is defined in terms of representations of the Lie algebra sl(r, C((z))).

On Generalized Theta Functions

We obtain a characterization of theta functions of Jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties (p.p.a.v.). We first give a characterization in terms of finite dimensional orbits for a suitable action in the Sato Grassmannian. Secondly, the introduction of formal Baker-Akhiezer functions and formal T -functions attached to a p.p.a.v. (for the multipuncture case) allows us to characterize, in terms of bilinear identities, those Baker-Akhiezer functions that are Baker-Akhiezer functions of Jacobians of curves with automorphisms. Further, in the case of automorphisms with fixed points, we rewrite the previous result as a hierarchy of partial differential equations for the τ-function of a p.p.a.v. Finally, since Baker-Akhiezer and T functions are written in terms of theta functions, these results give rise to characterizations of p.p.a.v. in terms of their theta functions.

It is known that holomorphic sections of an ample line bundle $L$ (and its tensor power $L^k$) on an Abelian variety $A$ are given by theta functions. Moreover, a natural basis of the space of holomorphic sections is related to a certain Lagrangian fibration of $A$. We study projective embeddings of $A$ given by the basis for $L^k$, and show that moment maps of toric actions on the ambient projective spaces, restricted to $A$, approximate the Lagrangian fibration of $A$ for large $k$. The case of Kummer variety is also discussed.

§My topic arises out of my efforts to respond to a question raised last summer by W. L. Hoyt at the AMS Summer Institute on Theta Functions: How can theta nullwerte be used to describe Hilbert modular surfaces, or, more generally, Humbert surfaces, as subvarieties of the variety of moduli of abelian surfaces (e. g., with principal polarization)? I want to show that an easy very slight re-packaging of the presentation by Mumford [4] of the subject of Riemann’s relations, which are general quartic relations satisfied by the homogeneous coordinates (for suitable projective embeddings) of the moduli of all abelian varieties, leads to special quadratic relations for these homogeneous coordinates when the moduli represent abelian varieties that are special in a certain sense.