[Paper Review] Exterior Powers of Barsotti-Tate Groups

Mohammad Hadi Hedayatzadeh|arXiv (Cornell University)|Sep 13, 2010

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes the existence and base-change compatibility of exterior powers for $π$-divisible ${\cal O}$-modules and $p$-divisible groups of dimension at most 1 over various bases, including fields and locally Noetherian schemes. The key contribution is a universal construction of exterior powers that preserves height and dimension, with explicit formulas: $\operatorname{ht}(\bigwedge^r G) = \binom{h}{r}$ and $\dim(\bigwedge^r G) = \binom{d-1}{r-1}$ for a $p$-divisible group $G$ of height $h$ and dimension $d \leq 1$. The results generalize classical constructions in arithmetic geometry and provide a framework for studying Weil pairings and Dieudonné modules.

ABSTRACT

Let $ \CO $ be the ring of integers of a non-Archimedean local field and $ π$ a fixed uniformizer of $ \CO $. We establish three main results. The first one states that the exterior powers of a $ π$-divisible $ \CO $-module scheme of dimension at most 1 over a field exist and commute with algebraic field extensions. The second one states that the exterior powers of a $p$-divisible group of dimension at most 1 over arbitrary base exist and commute with arbitrary base change. The third one states that when $ \CO $ has characteristic zero, then the exterior powers of $ π$-divisible groups with scalar $ \CO $-action and dimension at most 1 over a locally Noetherian base scheme exist and commute with arbitrary base change. We also calculate the height and dimension of the exterior powers in terms of the height of the given $p$-divisible group or $ π$-divisible $ \CO $-module scheme.

Motivation & Objective

• To establish the existence of exterior powers for $\pi$-divisible ${\cal O}$-modules of dimension at most 1 over fields.
• To extend this construction to $p$-divisible groups over arbitrary base schemes, ensuring compatibility with arbitrary base change.
• To generalize the theory to $\pi$-divisible ${\cal O}$-modules with scalar ${\cal O}$-action over locally Noetherian bases when ${\cal O}$ has characteristic zero.
• To compute the height and dimension of the exterior powers in terms of the original group's invariants.
• To provide a categorical universal property for exterior powers in the context of $R$-module schemes and $p$-divisible groups.

Proposed method

• Formalizing the universal property of exterior powers in the category of $R$-module schemes, using alternating $R$-multilinear morphisms.
• Applying $R$-multilinear Dieudonné theory and display theory to translate the problem into module-theoretic language.
• Using base change and Weil restriction techniques to prove compatibility of exterior powers with arbitrary base change.
• Constructing exterior powers via the universal property and verifying it via the associated Dieudonné modules and displays.
• Leveraging the theory of ramified displays and multilinear morphisms to handle the case of $\pi$-divisible ${\cal O}$-modules.
• Verifying the construction via Cartier duality and comparison with known examples, such as the Weil pairing on elliptic curves.

Experimental results

Research questions

• RQ1 Do exterior powers of $\pi$-divisible ${\cal O}$-modules of dimension at most 1 exist over fields, and do they commute with algebraic field extensions?
• RQ2 Can exterior powers of $p$-divisible groups of dimension at most 1 be constructed over arbitrary base schemes, and are they compatible with arbitrary base change?
• RQ3 What is the behavior of exterior powers when the base ring ${\cal O}$ has characteristic zero and the group carries a scalar ${\cal O}$-action?
• RQ4 How do the height and dimension of the exterior power $\bigwedge^r G$ relate to those of the original $p$-divisible group $G$?
• RQ5 Is the Weil pairing on an elliptic curve realized as the universal alternating morphism via the second exterior power of its $p$-divisible group?

Key findings

• The $r$-th exterior power $\bigwedge^r G$ of a $p$-divisible group $G$ of height $h$ and dimension $d \leq 1$ exists and has height $\binom{h}{r}$ and dimension $\binom{d-1}{r-1}$.
• For a $\pi$-divisible ${\cal O}$-module of dimension at most 1 over a field, the exterior powers exist and commute with algebraic field extensions.
• Over an arbitrary base scheme, the exterior powers of $p$-divisible groups of dimension at most 1 exist and are compatible with arbitrary base change.
• When ${\cal O}$ has characteristic zero, the exterior powers of $\pi$-divisible ${\cal O}$-modules with scalar action and dimension $\leq 1$ over locally Noetherian bases exist and commute with arbitrary base change.
• The second exterior power of the $p$-divisible group of an elliptic scheme is isomorphic to $\mu_{p^\infty}$, and the Weil pairing realizes this as the universal alternating morphism.
• In the case of a $p$-divisible group with slopes $0$ and $\frac{1}{n-m}$, the exterior power $\bigwedge^r G$ inherits a slope decomposition with explicit multiplicities.