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Global $\mathbb{A}^1$ degrees of covering maps between modular curves

Hyun Jong Kim, Sun Woo Park

Abstract

Given a projective smooth curve $X$ over any field $k$, we discuss two notions of global $\mathbb{A}^1$ degree of a finite morphism of smooth curves $f: X \to \mathbb{P}^1_k$ satisfying certain conditions. One originates from computing the Euler number of the pullback of the line bundle $\mathscr{O}_{\mathbb{P}^1}(1)$ as a generalization of Kass and Wickelgren's construction of Euler numbers. The other originates from the construction of global $\mathbb{A}^1$ degree of morphisms of projective curves by Kass, Levine, Solomon, and Wickelgren as a generalization of Morel's construction of $\mathbb{A}^1$-Brouwer degree of a morphism $f: \mathbb{P}^1_k \to \mathbb{P}^1_k$. We prove that under certain conditions on $N$, both notions of global $\mathbb{A}^1$ degrees of covering maps between modular curves $X_0(N) \to X(1)$, $X_1(N) \to X(1)$, and $X(N) \to X(1)$ agree to be equal to sums of hyperbolic elements $\langle 1 \rangle + \langle -1 \rangle$ in the Grothendieck-Witt ring $\mathrm{GW}(k)$ for any field $k$ whose characteristic is coprime to $N$ and the pullback of $\mathscr{O}_{\mathbb{P}^1}(1)$ is relatively oriented.

Global $\mathbb{A}^1$ degrees of covering maps between modular curves

Abstract

Given a projective smooth curve over any field , we discuss two notions of global degree of a finite morphism of smooth curves satisfying certain conditions. One originates from computing the Euler number of the pullback of the line bundle as a generalization of Kass and Wickelgren's construction of Euler numbers. The other originates from the construction of global degree of morphisms of projective curves by Kass, Levine, Solomon, and Wickelgren as a generalization of Morel's construction of -Brouwer degree of a morphism . We prove that under certain conditions on , both notions of global degrees of covering maps between modular curves , , and agree to be equal to sums of hyperbolic elements in the Grothendieck-Witt ring for any field whose characteristic is coprime to and the pullback of is relatively oriented.

Paper Structure

This paper contains 11 sections, 28 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. A1 homotopy theory
  4. A1 degrees of finite morphisms over projective spaces
  5. Euler numbers of vector bundles over smooth schemes
  6. Two variants of global A1 degrees of finite maps into the projective line
  7. Defining Euler numbers of finite maps into the projective line
  8. The Euler number of a finite map into An.
  9. An explicit method to compute local indices and global A1 degrees
  10. Covering maps of modular curves
  11. Future directions

Key Result

Theorem 1

Let $C$ be a smooth projective curve over a field $k$. Suppose that $\pi: C \to \mathbb{P}^1_k$ is a finite morphism satisfying some conditions specified in Theorem thm:two_notions_agree. If the pullback of line bundles $\pi^* \mathscr{O}_{\mathbb{P}^1}(1)$ and $\pi^* \mathscr{O}_{\mathbb{P}^1}(2)$

Theorems & Definitions (76)

  • Definition : Global Euler $\mathbb{A}^1$ degree, Definition \ref{['naiveEulerequiv']}
  • Definition 1.1: Global $\mathbb{A}^1$-degree
  • Theorem : Theorem \ref{['thm:two_notions_agree']}
  • Theorem : Theorem \ref{['mainA']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 66 more