Variation of height in an isogeny class over a function field

TL;DR

The paper investigates how differential and modular heights of abelian varieties over function fields vary along isogeny classes, introducing FV-isogenies and Frobenius height to capture characteristic-$p$ phenomena. It proves sharp height variation formulas: in characteristic 0 or when the degree is coprime to the characteristic, $h_{\mathsf{diff}}$ is preserved; in positive characteristic for FV-isogenies of type $(e,f)$, $h_{\mathsf{diff}}(B)=p^{e-f}h_{\mathsf{diff}}(A)$ for semi-stable cases, and more generally provides optimal bounds via Frobenius height in the ordinary case. A central achievement is a function-field parallelogram inequality for $h_{\mathsf{diff}}$, mirroring Remond’s number-field result and yielding subadditivity for heights in short exact sequences. The work also connects $h_{\mathsf{mod}}$ to $h_{\mathsf{st}}$ via Moret-Bailly’s Formule Clé, deriving corollaries on heights of subvarieties and quotients, and highlighting new height phenomena in higher dimensions especially in characteristic $p$. Overall, the results illuminate the nuances of height variation under isogenies in the function-field setting and provide tools for further Diophantine and Arakelov-type applications.

Abstract

We give optimal estimates on the variation of the differential and modular heights within an isogeny class of abelian varieties defined over the function field of a curve (in any characteristic). We also prove a parallelogram inequality for abelian varieties in this context, and deduce corollaries of these results.

Theorems & Definitions (134)

• Theorem
• Theorem A
• Corollary A.1
• Theorem B
• Corollary B.1
• Corollary B.2
• Proposition 1.1
• proof
• Definition 1.2
• Lemma 1.3