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Anharmonic Solutions to the Riccati equation and elliptic modular functions

Ahmed Sebbar, Oumar Wone

Abstract

We study algebraic solutions of the Riccati equation over the field of rational functions $\mathbb C(t)$, and over the elliptic function field $\mathbb C(\wp,\wp^\prime)$.

Anharmonic Solutions to the Riccati equation and elliptic modular functions

Abstract

We study algebraic solutions of the Riccati equation over the field of rational functions , and over the elliptic function field .

Paper Structure

This paper contains 9 sections, 15 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Algebraic solutions of the Riccati equation on $\mathbb {C}(t)$ and the cross-ratio
  3. Degree of the anharmonics
  4. Riccati equation and Darboux-Halphen Systems
  5. $e_1, e_2, e_3$ and the Darboux-Halphen system
  6. The degree three polynomial for the Riccati equation
  7. Algebraic solutions of the Riccati equation on ${\mathbb{C}(\wp(z),\wp^\prime(z))}$: invariant theory of binary forms
  8. Link with Hypergeometric Functions and Conclusion
  9. Remarks and Questions

Key Result

Theorem 1

Let be an integer $n\in\mathbb N\,,n\geqslant4$. Let $H\subset PGL(2,\mathbb {C})$ be a finite subgroup. We consider the equation a Riccati equation having algebraic solutions of degree $n$ and of Galois group $H$. There exists a polynomial $F_{n,H}(X,Y)$ in two variables and with constant coefficients ($\in\mathbb {C}$) such that for every algebraic solution $u$ of ($\star$) with minimal polynom

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Proposition 4
  • Lemma 5
  • proof : Proof of Theorem 1.1
  • Proposition 6
  • proof
  • Example 7
  • ...and 21 more