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

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 $f:X\to {\mathbb{P}}_k^1$ satisfying certain conditions. One originates from computing the Euler number of the pullback of the line bundle ${\mathcal{O}}_{{\mathbb{P}}^1}(1)$ 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 $f:{\mathbb{P}}_k^1\to {\mathbb{P}}_k^1$. 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 $⟨1⟩+⟨-1⟩$ in the Grothendieck-Witt ring $\mathrm{GW}(k)$ for any field $k$ whose characteristic is coprime to $N$ and the pullback of ${\mathcal{O}}_{{\mathbb{P}}^1}(1)$ is relatively oriented.

35 pages. Modified various statements to more precisely speak of "relatively oriented" maps or vector bundles instead of "relatively orientable" maps or vector bundles where appropriate --- the former phrasing suggests that a relative orientation is fixed. Additional minor edits