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

TL;DR

This work extends the classification of abelian varieties with real multiplication over finite fields by developing a Deligne-module framework for RM varieties on Hilbert moduli spaces under a mild Newton polygon assumption and a totally split prime $p$. It proves that, after lifting to characteristic zero and fixing CM-type ambiguities, RM abelian varieties fall into equivalence classes with Deligne modules, with precise ramification-dependent decompositions into 2^a or 2^{a−k} subcategories where functors yield category equivalences. The paper then derives sharp asymptotic bounds for the sizes of isogeny classes, showing $N(A,q^n)=q^{\frac{n}{2}(g-a+o(1))}$ for all but finitely many $n$, by comparing endomorphism rings to order-class groups and their over-orders; these bounds are reconciled via discriminant and polarization arguments. Overall, the results generalize prior work on simple and ordinary RM abelian varieties to the non-simple RM setting and provide a robust framework for understanding isogeny sizes through canonical lifts and Deligne-module techniques.

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.

Theorems & Definitions (54)

• Definition 1.1
• Definition 1.2
• Remark 1.1
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• Proposition 3.2
• proof
• Lemma 3.3
• Theorem 3.4