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A note on the inverse problem for finite differential Galois groups

Camilo Sanabria Malagón

Abstract

In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by M. van der Put, C. Sanabria and J.Top [19]. As an application, we show that there is no hypergeometric equation with differential Galois group isomorphic to $H_{72}^{SL_3}$, thus completing Beuker and Heckman's answer [4] to the question of which irreducible finite subgroup of $SL_3(\mathbb{C})$ are the monodromy of a hypergeometric equation.

A note on the inverse problem for finite differential Galois groups

Abstract

In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by M. van der Put, C. Sanabria and J.Top [19]. As an application, we show that there is no hypergeometric equation with differential Galois group isomorphic to , thus completing Beuker and Heckman's answer [4] to the question of which irreducible finite subgroup of are the monodromy of a hypergeometric equation.
Paper Structure (6 sections, 2 theorems, 39 equations, 1 figure)

This paper contains 6 sections, 2 theorems, 39 equations, 1 figure.

Table of Contents

  1. Preliminary
  2. Space of orbits
  3. Schwarz maps
  4. Algebraic Picard-Vessiot theory
  5. Schwarz maps for invariant curves
  6. The group $H_{72}^{SL_3}$

Key Result

Theorem 1

[Compoint's theorem for the algebraic case SANABRIA2017VANDERPUT2020] Let $L(y)=0$ be an irreducible linear ordinary differential equation. If the differential Galois group of $L(y)=0$ is finite, then the ideal $I$ is generated by the $G$-invariants contained in it. In particular, if $P_1,\ldots,P_N where $f_i=\Phi(P_i)$, $i=1,\ldots,N$.

Figures (1)

  • Figure 1: Diagram for Theorem \ref{['theo']}

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • proof