[Paper Review] p-adic monodromy of the universal deformation of an elementary Barsotti-Tate group

Yichao Tian|arXiv (Cornell University)|Aug 15, 2007

Advanced Algebra and Geometry2 citations

TL;DR

This paper proves that the p-adic monodromy representation of the fundamental group of the ordinary locus of the universal deformation of a connected, HW-cyclic Barsotti-Tate group over an algebraically closed field of characteristic p > 0 is surjective on the Tate module. The result generalizes Igusa’s theorem to a broader class of p-divisible groups, including all one-dimensional connected groups.

ABSTRACT

Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the algebraic local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $U\subset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $ ho$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $ ho$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivalent to that Oort's $a$-number of $G_0$ equals 1, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over $k$.

Motivation & Objective

• To extend Igusa’s theorem on monodromy surjectivity to a wider class of p-divisible groups beyond the ordinary case.
• To investigate the monodromy representation arising from the etale part of the universal deformation of a Barsotti-Tate group over its ordinary locus.
• To establish conditions under which the monodromy representation on the Tate module is surjective, focusing on the HW-cyclicity condition.
• To connect the monodromy behavior to the a-number of the Barsotti-Tate group, particularly when it equals 1.
• To provide a foundational result for the arithmetic monodromy of p-divisible groups in positive characteristic.

Proposed method

• Construct the algebraic local moduli space $ S $ for the Barsotti-Tate group $ G_0 $ in characteristic $ p $, parameterizing its deformations.
• Define $ U o S $ as the ordinary locus within $ S $, where the deformation $ G $ of $ G_0 $ becomes ordinary.
• Study the etale part of $ G $ over $ U $, which gives rise to a Galois representation $ ho $ of the fundamental group of $ U $ on the Tate module of $ G $.
• Use the condition that $ G_0 $ is connected and HW-cyclic—equivalent to Oort’s $ a $-number being 1—to analyze the structure of the monodromy representation.
• Apply techniques from p-adic Hodge theory and deformation theory to deduce the surjectivity of $ ho $, leveraging the geometry of the moduli space and the structure of the universal deformation.
• Establish the surjectivity of $ ho $ by analyzing the action on the Tate module and relating it to the p-adic monodromy of the universal family.

Experimental results

Research questions

• RQ1 Under what conditions is the p-adic monodromy representation of the fundamental group of the ordinary locus of a universal deformation of a Barsotti-Tate group surjective?
• RQ2 How does the HW-cyclicity condition on a Barsotti-Tate group relate to the surjectivity of its monodromy representation?
• RQ3 To what extent does Igusa’s classical result on monodromy surjectivity extend to non-ordinary, connected p-divisible groups?
• RQ4 What is the role of the a-number in determining the monodromy behavior of the universal deformation?
• RQ5 Can the monodromy representation on the Tate module be fully described for one-dimensional connected Barsotti-Tate groups in characteristic p?

Key findings

• The monodromy representation $ ho $ of the fundamental group of the ordinary locus $ U $ on the Tate module of the universal deformation $ G $ is surjective when $ G_0 $ is connected and HW-cyclic.
• The condition that $ G_0 $ is HW-cyclic is equivalent to Oort’s $ a $-number being equal to 1, which holds for all connected one-dimensional Barsotti-Tate groups over $ k $.
• The surjectivity of $ ho $ is established through a geometric analysis of the universal deformation over the moduli space $ S $ and its ordinary locus $ U $.
• The result generalizes Igusa’s theorem on monodromy surjectivity from the ordinary case to a broader class of p-divisible groups.
• The monodromy action on the Tate module captures the full Galois symmetry in the deformation space under the given conditions.
• The construction provides a uniform framework for studying monodromy in the context of p-adic Hodge theory and moduli of p-divisible groups in positive characteristic.