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Finiteness of Heights in Isogeny Classes of Motives with Semistable Reduction

Alice Lin

TL;DR

The work proves a finiteness property for the Kato–Koshikawa height in isogeny classes of motives with semistable reduction, assuming the adelic Mumford–Tate conjecture and related hypotheses. The authors develop a comprehensive framework using Breuil–Kisin–Fargues modules, Nygaard filtrations, and de Rham lattices to study height changes under isogenies, reducing to a slope analysis of BKF modules and their Galois/linear-group actions. They treat both fixed-prime and varying-prime scenarios: in the fixed-prime case they bound the limiting slope to show finiteness of heights, while in the varying-prime case they leverage the adelic MT conjecture and FL-type theory to control slopes across infinitely many primes. The results extend Kisin–Mocz finiteness from abelian varieties to motives, offering a robust approach to finiteness in Isogeny classes with semistable reduction and impacting the study of Shimura varieties and motives in arithmetic geometry.

Abstract

Using integral $p$-adic Hodge theory, Kato and Koshikawa define a generalization of the Faltings height of an abelian variety to motives defined over a number field. Assuming the adelic Mumford-Tate conjecture, we prove a finiteness property for heights in the isogeny class of a motive, where the isogenous motives are not required to be defined over the same number field. This expands on a result of Kisin and Mocz for the Faltings height in isogeny classes of abelian varieties.

Finiteness of Heights in Isogeny Classes of Motives with Semistable Reduction

TL;DR

The work proves a finiteness property for the Kato–Koshikawa height in isogeny classes of motives with semistable reduction, assuming the adelic Mumford–Tate conjecture and related hypotheses. The authors develop a comprehensive framework using Breuil–Kisin–Fargues modules, Nygaard filtrations, and de Rham lattices to study height changes under isogenies, reducing to a slope analysis of BKF modules and their Galois/linear-group actions. They treat both fixed-prime and varying-prime scenarios: in the fixed-prime case they bound the limiting slope to show finiteness of heights, while in the varying-prime case they leverage the adelic MT conjecture and FL-type theory to control slopes across infinitely many primes. The results extend Kisin–Mocz finiteness from abelian varieties to motives, offering a robust approach to finiteness in Isogeny classes with semistable reduction and impacting the study of Shimura varieties and motives in arithmetic geometry.

Abstract

Using integral -adic Hodge theory, Kato and Koshikawa define a generalization of the Faltings height of an abelian variety to motives defined over a number field. Assuming the adelic Mumford-Tate conjecture, we prove a finiteness property for heights in the isogeny class of a motive, where the isogenous motives are not required to be defined over the same number field. This expands on a result of Kisin and Mocz for the Faltings height in isogeny classes of abelian varieties.

Paper Structure

This paper contains 47 sections, 42 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Fixed prime case
  4. Varying prime case
  5. Organization of the paper
  6. Acknowledgements
  7. Notation
  8. Number fields, local fields, and Galois groups
  9. $p$-adic Hodge theory
  10. Miscellaneous
  11. Breuil--Kisin--Fargues modules
  12. Breuil--Kisin modules
  13. Breuil--Kisin--Fargues modules and Fargues' equivalence
  14. Base-changing from BK modules to BKF modules
  15. Slope of torsion BKF modules
  16. ...and 32 more sections

Key Result

Theorem 1.1

Let $A$ be an abelian variety defined over a number field. Assume that the Mumford--Tate conjecture holds for $A$. Then for any $c > 0$, the set contains finitely many isomorphism classes of abelian varieties over $\overline{\mathbb{Q}}$.

Theorems & Definitions (112)

  • Theorem 1.1: kisinmocznorthcott
  • Theorem 1
  • Theorem 2
  • Remark
  • Proposition 1.3: § \ref{['subsec:fixedprimeheightbound']}
  • Proposition 1.4: \ref{['cor:ptorsionheightbound']}
  • Definition 3.1.1
  • Theorem 3.1.3: kisin2006crystalline, liu2008lattices Lemma 3.4.5, kisin2010integral Theorem 1.2.1(2)
  • Definition 3.2.1: BMS2018integral, Definition 4.22
  • Theorem 3.2.3: scholzeweinstein2020berkeley, Theorem 14.1.1, originally due to Fargues
  • ...and 102 more