# Geometry and arithmetic associated to Appell hypergeometric partial differential equations

@article{Yang2003GeometryAA, title={Geometry and arithmetic associated to Appell hypergeometric partial differential equations}, author={Lei Yang}, journal={arXiv: Number Theory}, year={2003} }

In this paper, we study the monodromy of Appell hypergeometric partial differential equations, which lead us to find four derivatives which are associated to the group GL(3). Our four derivatives have the remarkable properties. We find that Appell hypergeometric partial differential equations can be reduced to four nonlinear partial differential equations by the use of our four derivatives. We investigate these four nonlinear partial differential equations. We are interested in some particular…

## 6 Citations

Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations

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We, briefly, recall the Fuchs–Painlevé elliptic representation of Painlevé VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the…

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We, briefly, recall the Fuchs-Painleve elliptic representation of Painleve VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the…

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In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry),…

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Modular equations for Appell’s F1 is developed from the work of Koike et al. A simpler derivation of their modular equations for Picard modular functions is found, and further modular equation are…

## References

SHOWING 1-10 OF 54 REFERENCES

Linear Differential Equations and Group Theory from Riemann to Poincare

- Mathematics
- 1986

The origins of the theory of modular and
automorphic functions are found in the work of Legendre, Gauss, Jacobi, and Kummer on elliptic functions and the hypergeometric equation.
Riemann's work on…

THE PAINLEVE PROPERTY FOR PARTIAL DIFFERENTIAL EQUATIONS. II. BACKLUND TRANSFORMATION, LAX PAIRS, AND THE SCHWARZIAN DERIVATIVE

- Mathematics
- 1983

In this paper we investigate the Painleve property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order…

The arithmetic of automorphic forms with respect to a unitary group

- Mathematics
- 1978

utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms. One expresses a…

ARITHMETIC OF UNITARY GROUPS

- Mathematics
- 1964

The purpose of this paper is to develop the theory of elementary divisors, to prove the approximation theorem, and to determine the class number for the following two types of algebraic groups: (i)…

Commensurabilities among lattices in PU(1,n)

- Mathematics
- 1993

The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n-variables. These are treated as an (n+1)…

Bäcklund transformation and the Painlevé property

- Mathematics
- 1986

When a differential equation possesses the Painleve property it is possible (for specific equations) to define a Backlund transformation (by truncating an expansion about the ‘‘singular’’ manifold at…

Quantum field theory, Grassmannians, and algebraic curves

- Mathematics
- 1988

This paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite Grassmannians, and in part to pointing out the existence of a close analogy between…

Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés abéliennes

- Mathematics
- 1993

The monodromy groups of the Appell-Lauricella functions Fi in several variables can be discontinuous, but this does not in general entail that they be arithmetically defined ([DM], [M]). Nonetheless,…

Introduction to the arithmetic theory of automorphic functions

- Mathematics
- 1971

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary…

Some Contemporary Problems with Origins in the Jugendtraum

- Mathematics
- 2001

The twelfth problem of Hilbert reminds us, although the reminder should be unnecessary, of the blood relationship of three subjects which have since undergone often separate developments. The first…