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Unlikely intersection in higher-dimensional formal groups

Mabud Ali Sarkar, Absos Ali Shaikh

TL;DR

This work addresses whether higher-dimensional formal groups can be recovered from their torsion points and whether abelian varieties can be distinguished up to isogeny by $p$-power torsion data. It extends Berger's 1-dimensional result to $d$-dimensional simple full-CM formal groups over $\mathbb{Z}_p$ by analyzing the $p$-adic Tate module as a crystalline Galois representation and employing endomorphism-commutativity via logarithms to show that infinite torsion intersection forces equality of the groups. The main contribution is Theorem $t3.6$: if $F$ and $G$ are as above and $\text{Tors}(F)\cap\text{Tors}(G)$ is infinite, then $F=G$, which yields a practical criterion: two abelian varieties with CM whose $p$-power torsion intersect infinitely often are isogenous. This result connects $p$-adic Hodge theory, Lubin–Tate-type structures, and endomorphism dynamics to concrete isogeny-detection, with potential implications for moduli of abelian varieties and their $L$-functions.

Abstract

In this work, we identify a certain family of higher-dimensional formal groups over the ring of $p$-adic integers such that any two formal groups in that class coincide if they share infinitely many torsion points. As a useful application, we provide a sufficient condition for two abelian varieties to be isogenous just by looking at their $p$-power torsion points.

Unlikely intersection in higher-dimensional formal groups

TL;DR

This work addresses whether higher-dimensional formal groups can be recovered from their torsion points and whether abelian varieties can be distinguished up to isogeny by -power torsion data. It extends Berger's 1-dimensional result to -dimensional simple full-CM formal groups over by analyzing the -adic Tate module as a crystalline Galois representation and employing endomorphism-commutativity via logarithms to show that infinite torsion intersection forces equality of the groups. The main contribution is Theorem : if and are as above and is infinite, then , which yields a practical criterion: two abelian varieties with CM whose -power torsion intersect infinitely often are isogenous. This result connects -adic Hodge theory, Lubin–Tate-type structures, and endomorphism dynamics to concrete isogeny-detection, with potential implications for moduli of abelian varieties and their -functions.

Abstract

In this work, we identify a certain family of higher-dimensional formal groups over the ring of -adic integers such that any two formal groups in that class coincide if they share infinitely many torsion points. As a useful application, we provide a sufficient condition for two abelian varieties to be isogenous just by looking at their -power torsion points.
Paper Structure (7 sections, 12 theorems, 37 equations)

This paper contains 7 sections, 12 theorems, 37 equations.

Table of Contents

  1. Introduction and Motivation
  2. Preliminary
  3. Unlikely intersection in higher-dimensional formal groups
  4. Alternative proof of Proposition \ref{['pp3.9']}
  5. An application to isogeneity of abelian varieties
  6. Conclusion and Future Direction
  7. Number of points in an algebraic set

Key Result

Theorem 1.1

LB1 If $F$ and $G$ are one-dimensional formal groups over $\mathcal{O}_K$ such that $\text{Tors}(F) \cap \text{Tors}(G)$ is infinite, then $F=G$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 37 more