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Fields with no everywhere good abelian varieties

This paper extends the methods of Fontaine, Abrashkin, and Schoof to establish criteria for identifying number fields that admit no non-zero abelian varieties with everywhere good reduction, and under the Generalized Riemann Hypothesis, applies these criteria to find 24,744 such fields with degrees up to 16.

Original authors: Armand Brumer, Kenneth Kramer

Published 2026-03-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Armand Brumer, Kenneth Kramer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Perfect Neighborhood" Problem

Imagine you are an architect trying to build a special kind of house called an Abelian Variety. These aren't ordinary houses; they are complex mathematical structures that live inside a specific "neighborhood" called a Number Field (think of this as a specific type of mathematical universe).

Every house has a "foundation" that needs to be stable. In math, we call this reduction.

Good Reduction: The house stands perfectly firm at every single point in the neighborhood. No cracks, no wobbles, no matter where you look.
Bad Reduction: The house has a crack or a wobble at a specific point (a "prime" or a "place").

The Question: Can you find a neighborhood (a Number Field) where it is impossible to build even one non-empty house that is perfectly stable everywhere?

The authors call such a neighborhood a "Fontaine Field." If a field is a Fontaine Field, it's like a cursed land where the laws of physics simply won't allow a perfect, uncracked house to exist.

The Previous State of Affairs

Before this paper, mathematicians knew of only about 16 specific neighborhoods where this "curse" existed. It was like finding 16 islands where no perfect sandcastles could be built. The big mystery was: Are there more? How many? And how do we find them?

The Authors' New Strategy: The "Security Check"

Brumer and Kramer developed a new, super-strict security check to prove that a neighborhood is a Fontaine Field. They didn't just look at the houses; they looked at the blueprints and the laws of the neighborhood.

Here is how their method works, broken down into simple steps:

1. The "2-Extension" Filter (The Gatekeeper)

Imagine the neighborhood has a special gate. To get a perfect house, the blueprint must pass through this gate. The authors realized that if the neighborhood has a specific property (related to how "twisted" the land is), the gate only allows blueprints that are "powers of 2" to pass through.

Analogy: Think of it like a bouncer at a club who only lets in people wearing shirts with a specific number of buttons. If the neighborhood is "twisted" enough, the bouncer only lets in "2-power" blueprints.

2. The "Ramification" Speed Limit (The Traffic Cop)

In math, "ramification" is like a traffic jam or a sudden change in the road. The authors used a famous "speed limit" rule (from mathematicians Fontaine and Abrashkin).

The Rule: If a house is perfect everywhere, the traffic jams in its blueprint cannot get too severe.
The Catch: The authors calculated that in many neighborhoods, the "traffic jams" required to build a perfect house would have to be worse than the speed limit allows. Therefore, the house cannot be built.

3. The "Group Theory" Puzzle (The Lego Blocks)

To prove the house can't be built, they broke the blueprint down into tiny Lego blocks (called Group Schemes).

They asked: "Can we stack these Lego blocks to make a house?"
They found that in certain neighborhoods, the rules of the land (specifically, the lack of certain "quadratic extensions" or "side roads") prevent the Lego blocks from snapping together in the right way.
The Metaphor: Imagine trying to build a tower with Legos, but the instructions say, "You can only use red blocks, but the box only contains blue ones." If you can't get the right blocks, the tower (the Abelian Variety) collapses.

The Results: A Massive Discovery

Using a supercomputer and assuming a famous mathematical guess called the GRH (Generalized Riemann Hypothesis—which is like assuming the weather forecast is 99% accurate), they ran this security check on thousands of neighborhoods.

The Outcome:
They found 24,744 new neighborhoods (Number Fields) where it is impossible to build a perfect house.

They looked at neighborhoods of different sizes (degrees from 2 up to 16).
They checked fields with 1, 2, or 3 "traffic jams" (primes over 2).

Why Does This Matter?

You might ask, "Who cares if we can't build a perfect house in a math universe?"

Mapping the Landscape: Before this, we only knew of a few "cursed" islands. Now we have a map of thousands. It helps mathematicians understand the "geography" of numbers.
Testing Limits: It pushes the boundaries of what we know about how numbers behave. It's like testing how much weight a bridge can hold before it breaks.
The "Schoof Pairs": The paper also touches on a related question: Are there neighborhoods where only a few types of houses can be built? This helps classify the "population" of mathematical structures.

Summary Analogy

Imagine you are a city planner.

Old Knowledge: You knew that in 16 specific cities, you couldn't build a skyscraper that didn't have a crack in the foundation.
New Knowledge: Brumer and Kramer invented a new way to test the soil. They ran their test on thousands of cities and found 24,744 more cities where the soil is so unstable that no perfect skyscraper can ever be built.

They didn't just find the cities; they explained why the soil is unstable (using the "traffic jams" and "Lego block" rules), giving future architects a better understanding of the mathematical world.

1. Problem Statement and Context

The central problem addressed is the non-existence of non-zero abelian varieties with everywhere good reduction over specific number fields $K$ .

Definition: A number field $K$ is called a Fontaine field if there exists no non-zero abelian variety over $K$ with everywhere good reduction.
Background: Fontaine (1985) and Abrashkin (1990) independently proved that $\mathbb{Q}$ is a Fontaine field. Schoof later identified specific cyclotomic and quadratic fields as Fontaine fields.
The Gap: Prior to this work, only 16 specific number fields were known to be Fontaine fields. The authors aim to extend these results to a much broader class of number fields, specifically those of degree $n \le 16$ , under the assumption of the Generalized Riemann Hypothesis (GRH).

2. Methodology

The authors synthesize tools from arithmetic geometry, group theory, and class field theory to construct criteria that force the non-existence of such varieties.

A. Theoretical Framework

Ramification Bounds (Fontaine-Abrashkin):
- They utilize bounds on higher ramification groups for finite flat group schemes of exponent $p^n$ over the ring of integers $\mathcal{O}_K$ .
- Specifically, for exponent 2, the upper ramification bound implies that the field of points $K(E)$ for such a group scheme $E$ is contained within a specific extension $F/K$ (the Abrashkin-Fontaine extension).
- The root discriminant of $F$ is bounded by $\delta_F < 4\delta_K$ .
Group Schemes and Filtrations:
- The authors analyze the structure of the 2-torsion $A[2^n]$ of an abelian variety $A$ .
- They define a prosaic abelian scheme as one where the 2-division field $K(A[2])$ is a 2-extension of $K$ .
- Using group schemes of order 2 (denoted $G_\rho$ ), they construct filtrations of $A[2^n]$ . The goal is to show that under certain conditions, the extension classes between these group schemes vanish or are highly constrained, forcing the abelian variety to be trivial ( $A=0$ ).
Group Theoretic Constraints:
- The paper employs deep group theory (Propositions 3.1–3.3) to analyze the Galois group $\Gamma = \text{Gal}(F/K)$ .
- They prove that if $\Gamma$ contains odd primes in its derived series, it must admit a dihedral quotient $D_p$ . By ensuring specific ray class groups are 2-groups, they eliminate these odd primes, forcing the maximal solvable extension to be a 2-extension.

B. Computational Strategy

The authors implement a rigorous filtering process to identify candidate fields $K$ satisfying Hypothesis K:

$K$ has odd discriminant.
The narrow class number $h^+_K = 1$ .
There are at most 3 primes lying above 2 in $K$ .

The Algorithm:

Data Source: Utilize the LMFDB (L-functions and Modular Forms Database) and Magma for number fields of degree up to 16.
Witness Fields ( $K_p$ ): For primes $p|2\mathcal{O}_K$ , they define a "witness field" $K_p$ (the maximal elementary 2-extension with specific conductor conditions).
Dichotomy Check: They verify one of two conditions:
- $D1(p)$ : $K_p = K$ .
- $D2(p)$ : $[K_p : K] = 2$ with $p$ inert, and no quadratic extension splits over $p$ with specific conductors.
Ray Class Group Verification: They compute ray class groups for quadratic extensions within the tower. If any ray class group contains an odd prime factor, the field is rejected. This ensures the Galois group of the relevant extensions is a 2-group.
Odlyzko Bounds: Under GRH, they apply Odlyzko's discriminant bounds. If the theoretical degree of the maximal extension $F$ exceeds the Odlyzko bound for the root discriminant of $K$ , they conclude $F$ must be a 2-extension, implying no non-trivial abelian varieties exist.

3. Key Contributions

Generalization of Criteria: The paper extends the methods of Fontaine, Abrashkin, and Schoof by introducing a "witness field" $K_p$ and utilizing group-theoretic implications of ramification bounds rather than just discriminant bounds.
Structural Analysis of Group Schemes: They provide a detailed classification of extension classes for group schemes of order 2 (Proposition 4.8, 4.11, 4.17), showing how to rearrange filtrations to isolate specific constituents ( $\mu_2$ , $\mathbb{Z}/2\mathbb{Z}$ , and $G_\rho$ ).
Non-Existence Theorems: They establish Theorems 6.4 and 6.5, which provide sufficient conditions (based on the behavior of primes over 2 and ray class groups) for a field to be a Fontaine field.
Large-Scale Verification: They successfully automate these checks for thousands of fields.

4. Results

Scale: The authors identified 24,744 distinct number fields of degree $n \le 16$ that are Fontaine fields.
Distribution:
- The count varies significantly by degree (e.g., 12 fields of degree 2, 22 of degree 11, 18,478 of degree 10).
- The results cover fields with 1, 2, or 3 primes lying above 2.
Comparison: Their results are compared with a preprint by Tchamitchian ([Tch]), who used Weil's explicit formulas. The authors note that their 24,744 fields largely overlap with Tchamitchian's findings, providing mutual support for their respective methods and assumptions (GRH vs. L-function conjectures).

5. Significance

Expansion of Known Examples: This work dramatically increases the known universe of Fontaine fields from 16 to nearly 25,000, providing a robust dataset for testing conjectures in arithmetic geometry.
Methodological Advancement: The paper demonstrates a powerful hybrid approach combining local ramification theory, global class field theory, and computational number theory. It moves beyond "ad hoc" proofs for specific fields to a systematic, algorithmic framework.
Implications for Abelian Varieties: The results suggest that "everywhere good reduction" is an extremely rare property for abelian varieties over number fields, reinforcing the intuition that such varieties are likely non-existent for most fields.
Foundation for Future Work: The criteria established (particularly regarding the structure of the maximal solvable extension and the role of witness fields) provide a blueprint for investigating higher degrees or relaxing the GRH assumption in future research.

In summary, Brumer and Kramer provide a definitive, computationally verified expansion of the theory of Fontaine fields, proving that for a vast array of number fields (up to degree 16), the existence of non-zero abelian varieties with everywhere good reduction is impossible.