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Galois representation of the product of two Drinfeld modules of generic characteristic

Lian Duan, Jiangxue Fang

Abstract

In this paper, we study the Galois representations attached to products of Drinfeld modules. As an analogue of Serre's classical result on the images of Galois representations associated with products of elliptic curves, we prove that for any finite set of primes, the image of the corresponding product representation is sufficiently large, in the sense that it is commensurable with a subgroup defined by a natural determinant condition. Our approach combines Pink's minimal model theory for compact subgroups of linear groups over local fields with explicit reciprocity laws for global function fields.

Galois representation of the product of two Drinfeld modules of generic characteristic

Abstract

In this paper, we study the Galois representations attached to products of Drinfeld modules. As an analogue of Serre's classical result on the images of Galois representations associated with products of elliptic curves, we prove that for any finite set of primes, the image of the corresponding product representation is sufficiently large, in the sense that it is commensurable with a subgroup defined by a natural determinant condition. Our approach combines Pink's minimal model theory for compact subgroups of linear groups over local fields with explicit reciprocity laws for global function fields.

Paper Structure

This paper contains 19 sections, 18 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Statement of the main theorem
  3. Structure of this paper
  4. Some simplifications
  5. Arithmetic of Drinfeld modules
  6. Tate modules
  7. Frobenius twist
  8. Specialization
  9. Adjoint Galois representation of two Drinfeld modules
  10. Minimal model theory of compact subgroups of linear algebraic groups
  11. Minimal model theory associated to Galois presentation of Drinfeld modules
  12. Adjoint representation of $\psi$ and $\psi'$ at two maximal ideals
  13. Adjoint representation of $\psi$ and $\psi'$ at finitely many maximal ideals
  14. Determinant representation of $\psi$ and $\psi'$
  15. Openness of the norm map of global fields
  16. ...and 4 more sections

Key Result

Theorem 1.1

PR There exists an open subgroup $N$ of $\Gamma_K$ such that $\rho(N)$ is open in $\prod\limits_{\mathfrak p\in{\rm Max}\,A}{\rm GL}_{A_\mathfrak p\otimes_A{\rm End}_{\overline K}(\phi)}(T_\mathfrak p\phi)$, which is endowed with the product topology.

Theorems & Definitions (34)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • ...and 24 more