[Paper Review] The filled Julia set of a Drinfeld module and uniform bounds for torsion

Patrick Ingram|arXiv (Cornell University)|Oct 10, 2012

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper establishes a dynamical systems framework for Drinfeld modules by introducing the filled Julia set and its component module over local and adelic fields. It proves that uniform bounds on torsion in Drinfeld modules follow from a conjecture on the structure of this component module, and verifies the conjecture for certain families, yielding explicit uniform torsion bounds in special cases.

ABSTRACT

If M is a Drinfeld module over a local function field L, we may view M as a dynamical system, and consider its filled Julia set J. If J^0 is the connected component of the identity, relative to the Berkovich topology, we give a characterisation of the component module J/J^0 which is analogous to the Kodaira-Neron characterisation of the special fibre of a Neron model of an elliptic curve over a non-archimedean field. In particular, if L is the fraction field of a discrete valuation ring, then the component module is finite, and moreover trivial in the case of good reduction. In the context of global function fields, the filled Julia set may be considered as an object over the ring of finite adeles. In this setting we formulate a conjecture about the structure of the (finite) component module which, if true, would imply Poonen's Uniform Boundedness Conjecture for torsion on Drinfeld modules of a given rank over a given global function field. Finally, we prove this conjecture for certain families of Drinfeld modules, obtaining uniform bounds on torsion in some special cases.

Motivation & Objective

• To establish a dynamical systems interpretation of Drinfeld modules using the filled Julia set and component module over local and global function fields.
• To formulate a conjecture on the structure of the adelic component module that implies Poonen's Uniform Boundedness Conjecture for torsion in Drinfeld modules.
• To prove the conjecture for specific families of Drinfeld modules, thereby establishing uniform torsion bounds in these cases.
• To draw an analogy with the Kodaira-Néron model for elliptic curves, providing a parallel structure for component modules in the Drinfeld module setting.
• To define a height-like invariant μ(φ, N, a) measuring the proportion of places where the component module is small, linking it to torsion bounds.

Proposed method

• Define the filled Julia set FJ(φ) as the maximal bounded submodule of φ(C_v) over the completion C_v of an algebraic closure of a local function field L_v.
• Introduce the connected component φ⁰(L_v) of the identity in the Berkovich topology, and study the quotient FJ(φ, L_v)/φ⁰(L_v), which forms a finite A-module.
• Use the Berkovich analytic space A¹_Berk to analyze the topology and structure of the filled Julia set and its component.
• Define a measure μ(φ, N, a) ∈ [0, 1] that quantifies the proportion of finite places (up to N exceptions) where the component module is annihilated by an ideal a.
• Prove that if μ(φ, N, a) ≥ 1/q, then the torsion subgroup φ^Tors(L) is uniformly bounded in size by a constant depending only on r, N, and a.
• Apply the theory of Drinfeld modules over global function fields by considering the adelic filled Julia set and using the component module structure to derive torsion bounds.

Experimental results

Research questions

• RQ1 Can the component module FJ(φ, L_v)/φ⁰(L_v) of a Drinfeld module over a local function field be characterized in a way analogous to the Kodaira-Néron model for elliptic curves?
• RQ2 Does a conjecture on the structure of the adelic component module imply Poonen’s Uniform Boundedness Conjecture for torsion in Drinfeld modules?
• RQ3 For which families of Drinfeld modules is the conjecture on the component module structure verified, and what uniform torsion bounds can be derived?
• RQ4 How does the invariant μ(φ, N, a) relate to the distribution of places where the component module is small, and how does it control torsion growth?
• RQ5 To what extent does the analogy with elliptic curves extend to the component module and height-based measures in the Drinfeld module setting?

Key findings

• The component module FJ(φ, L_v)/φ⁰(L_v) is a finite A-module, and is trivial in the case of good reduction.
• For any Drinfeld module φ over a global function field, the adelic component module is finite, and its structure governs torsion bounds.
• If μ(φ, N, a) ≥ 1/q, then the size of the torsion subgroup φ^Tors(L) is bounded by a constant depending only on the rank r, the ideal a, and the number N of bad reduction places.
• The conjecture on the component module structure implies Poonen’s Uniform Boundedness Conjecture for Drinfeld modules of a given rank over a given global function field.
• The conjecture is proven for certain families of Drinfeld modules, yielding explicit uniform torsion bounds in these cases.
• The method generalizes earlier results of Ghioca by using the filled Julia set and component module as central objects, extending the analogy with elliptic curves.