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Relation between Anderson Generating Functions and Weil Pairing

Chuangqiang Hu, Yixuan Ou-Yang

Abstract

The existence of the Weil pairing for Drinfeld modules was proved by van~der~Heiden using the Anderson $t$-motive. Papikian's note provided the explicit formula for the rank-two Weil pairing that avoids Anderson motives. Following this approach, Katen extended the formula to higher ranks. As Papikian observed, this method is more elementary than the approach using Anderson motives, but it is less conceptual. This paper is devoted to a new insight into Katen's formula motivated by the Moore determinant coming from Hamahata's tensor product of Drinfeld modules and the basis of torsion modules found by Maurischat and Perkins. We investigate the Weil operator, establish its connection with the remainder polynomial of Anderson generating functions modulo a fixed polynomial $\f$, and finally derive an extremely simple interpretation: the value of the rank-$r$ Weil pairing is essentially the specific coefficient in the Moore determinant of certain Anderson generating functions.

Relation between Anderson Generating Functions and Weil Pairing

Abstract

The existence of the Weil pairing for Drinfeld modules was proved by van~der~Heiden using the Anderson -motive. Papikian's note provided the explicit formula for the rank-two Weil pairing that avoids Anderson motives. Following this approach, Katen extended the formula to higher ranks. As Papikian observed, this method is more elementary than the approach using Anderson motives, but it is less conceptual. This paper is devoted to a new insight into Katen's formula motivated by the Moore determinant coming from Hamahata's tensor product of Drinfeld modules and the basis of torsion modules found by Maurischat and Perkins. We investigate the Weil operator, establish its connection with the remainder polynomial of Anderson generating functions modulo a fixed polynomial , and finally derive an extremely simple interpretation: the value of the rank- Weil pairing is essentially the specific coefficient in the Moore determinant of certain Anderson generating functions.

Paper Structure

This paper contains 33 sections, 32 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Weil Pairing of Drinfeld Modules
  3. Van der Heiden's Construction
  4. Explicit Formula
  5. Main Results
  6. Outline
  7. Notations
  8. Fields and Rings
  9. Drinfeld Modules
  10. Polynomials and Ideals
  11. Weil Operators
  12. Tate algebra
  13. Anderson Generating Functions
  14. Weil Pairing
  15. Weil Operator
  16. ...and 18 more sections

Key Result

Lemma 2.2

With the notations above, we have where $\eta_\mathfrak{f}^* = -\frac{1}{\mathfrak{f}} dt$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Graphs $G^{\gamma_i}$

Theorems & Definitions (73)

  • Definition 1.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 63 more