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

TL;DR

This paper compares two notions of global -degree for covering maps between modular curves, showing they agree and can be expressed as sums of hyperbolic elements in the Grothendieck-Witt ring under certain conditions.

Contribution

It establishes the equivalence of two definitions of global -degree for modular curve coverings and computes these degrees explicitly in terms of hyperbolic elements.

Findings

Both notions of global -degree agree for certain modular curve covers.

The degrees are expressed as sums of hyperbolic elements  +  in -ring.

Results hold over fields with characteristic coprime to N and under relative orientability.

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.