Finiteness for Étale Fundamental Groups of Néron Models

Technical Summary: Finiteness for Étale Fundamental Groups of Néron Models

Problem Statement
The paper investigates the structure and finiteness properties of the étale fundamental group of the Néron model of an abelian variety over a number field $K$. Specifically, it focuses on the "geometric" part of this group, denoted $\pi_{1}^{\text{et,geom}}(\mathcal{A})$, defined as the kernel of the map $\pi_{1}^{\text{et}}(\mathcal{A}) \to \pi_{1}^{\text{et}}(\text{Spec } \mathcal{O}_K)$. While previous results by Katz and Lang established the finiteness of the full fundamental group for abelian varieties over $\mathbb{Q}$, and Bost and Charles proved finiteness for integral regular models of elliptic curves, the finiteness of the geometric subgroup for general abelian varieties over arbitrary number fields remained an open question. The paper aims to prove that this subgroup is finite and to determine if its size can be uniformly bounded depending only on the dimension and the base field.

Methodology
The primary strategy involves analyzing the behavior of the Faltings height under finite étale covers of Néron models. The author leverages Faltings's isogeny formula, which relates the heights of isogenous abelian varieties to the degree of the isogeny. The core argument proceeds by contradiction: if the geometric fundamental group were infinite, one could construct a sequence of finite étale covers (and thus isogenies) of arbitrarily large degree. By Faltings's formula, such isogenies would produce abelian varieties with arbitrarily small Faltings heights. This contradicts the Northcott property for abelian varieties over number fields, which asserts that there are only finitely many isomorphism classes of abelian varieties with height bounded above a constant.

For the specific case of elliptic curves, the methodology is refined to establish uniformity. The author utilizes Merel's uniform torsion theorem, which bounds the size of the torsion subgroup of elliptic curves over a fixed number field. By showing that the kernel of a finite étale cover of a Néron model must be defined over the Hilbert class field of $K$, the problem is reduced to bounding torsion points over this specific extension.

In the case of elliptic curves over $\mathbb{Q}$, the paper employs a combination of arithmetic geometry and explicit computation. The author analyzes the discriminant of the elliptic curve under isogeny, showing that for a finite étale cover of degree $n$, the minimal discriminant of the cover must be the $n$-th power of the original discriminant. This condition, combined with Silverman's formula for the Faltings height and Mazur's torsion theorem, restricts the possible degrees of such covers to a small set of integers. The non-existence of covers of degree 7 is proven by interpreting the existence of such a curve as a rational point on a modular curve $X(1)$ that must simultaneously satisfy conditions derived from the equation $A^2 + B^3 = 1728C^7$ and the geometry of the modular curve $X_1(7)$. This involves constructing specific twists of $X(7)$ and performing local $p$-adic tests (using Sage) to show incompatibility.

Key Contributions and Results

1. General Finiteness (Theorem 1.1): The paper proves that for any abelian variety $A$ of dimension $g$ over a number field $K$, the geometric étale fundamental group $\pi_{1}^{\text{et,geom}}(\mathcal{A})$ of its Néron model is a finite group. This is achieved by demonstrating that the degrees of finite étale covers of the Néron model are bounded, preventing the existence of arbitrarily small Faltings heights.

2. Uniformity for Elliptic Curves (Theorem 1.3): For elliptic curves, the finiteness result is strengthened to a uniform bound. The paper proves that there exists a constant $C$, depending only on the number field $K$, such that $|\pi_{1}^{\text{et,geom}}(\mathcal{E})| \le C$ for any elliptic curve $E/K$. This relies on the fact that the kernel of the isogeny spreads out to a finite étale cover of $\text{Spec } \mathcal{O}_K$, implying the kernel points are defined over the Hilbert class field, where Merel's theorem applies.

3. Explicit Classification over $\mathbb{Q}$ (Theorem 1.4): For elliptic curves over $\mathbb{Q}$, the paper provides a complete list of possible étale fundamental groups for their Néron models. The group is isomorphic to $\mathbb{Z}/N\mathbb{Z}$ where $N \in \{1, 2, 3, 5\}$.

   • The case $N=7$ is ruled out. The proof shows that a degree 7 finite étale cover would require the minimal discriminant to be a seventh power and the existence of a rational 7-torsion point. By analyzing the modular curve $X(1)$ and the equation $A^2 + B^3 = 1728C^7$, the author demonstrates via local tests that no such rational points exist that correspond to valid elliptic curves with the required reduction properties.
   • The case $N=4$ is ruled out for curves with additive reduction by analyzing the component groups of the special fibers and using Silverman's height formula to show no such isogeny can exist.
   • Examples for $N=1, 2, 3, 5$ are provided using data from the LMFDB (L-functions and Modular Forms Database), confirming that all listed cases are realized.

Significance
The paper establishes a fundamental finiteness property for the geometric étale fundamental groups of Néron models, extending previous results from $\mathbb{Q}$ to arbitrary number fields and from integral regular models to the full Néron model. By proving that these groups are finite and uniformly bounded for elliptic curves, the work connects the arithmetic of isogenies, the geometry of Néron models, and the distribution of torsion points. The explicit classification over $\mathbb{Q}$ offers a concrete understanding of the possible symmetries of these models, ruling out specific group structures (like $\mathbb{Z}/7\mathbb{Z}$) through a synthesis of modular curve theory and Diophantine analysis. The results rely on standard tools in arithmetic geometry (Faltings height, Northcott property, Merel's theorem) but apply them in a novel context to resolve questions about the global structure of Néron models.