Abelian varieties with real multiplication: classification and isogeny classes over finite fields

TL;DR

This paper classifies specific points on Hilbert modular varieties over finite fields and uses this to estimate the size of isogeny classes, advancing understanding of abelian varieties with real multiplication.

Contribution

It offers a new classification of points on Hilbert modular varieties over finite fields and derives bounds on isogeny class sizes, linking geometric and arithmetic properties.

Findings

Classification of points on Hilbert modular varieties under Newton polygon assumptions

Estimates for the size of isogeny classes of abelian varieties with real multiplication

Enhanced understanding of the structure of abelian varieties over finite fields

Abstract

In this paper, we provide a classification of certain points on Hilbert modular varieties over finite fields under a mild assumption on Newton polygon. Furthermore, we use this characterization to prove estimates for the size of isogeny classes.