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Size of isogeny classes of abelian varieties of Lubin-Tate type

This paper establishes a lower bound for the size of isogeny classes of simple abelian varieties with commutative endomorphism rings in the Lubin-Tate case over finite fields and conjectures that this bound is sharp based on expected sizes within Newton strata.

Original authors: Tejasi Bhatnagar

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tejasi Bhatnagar

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

Imagine a vast, magical garden called the Newton Stratum. This isn't a garden of flowers, but of Abelian Varieties—complex, multi-dimensional shapes that live in a world of finite fields (think of a universe with a limited number of colors, like a pixelated video game).

In this garden, some shapes are "simple" (they can't be broken down into smaller, independent shapes), and they all share a secret code called a Newton Polygon. This polygon is like a fingerprint; it tells us the shape's basic "slope" and structure.

The Big Question: How Many Friends Does One Shape Have?

The paper asks a simple but tricky question: If you pick one specific shape in this garden, how many other shapes can you find that are isogenous to it?

Think of "isogenous" as a special kind of friendship. Two shapes are isogenous if they can be transformed into each other by a specific type of magic spell (an isogeny). They aren't identical twins, but they are close cousins who share the same DNA. The set of all these cousins is called an Isogeny Class.

The author, Tejas Bhatnagar, wants to know: How big is this family?

The "Lubin-Tate" Garden Patch

The paper focuses on a very specific, rare patch of the garden called the Lubin-Tate type.

The Rule: In this patch, the shapes have a very specific slope structure. If the garden has dimension $g$ (think of this as the number of "directions" the shape can wiggle), the slopes of the shapes in this patch are exactly $1/g$ and $(g-1)/g$ .
The Exclusion: The paper explicitly says, "We are not looking at the supersingular cases." If $g=2$ , the shape is supersingular, and the answer is boring (everyone is friends with everyone). So, the author sets the rule: $g > 2$ . We are only interested in the more complex, non-supersingular shapes.

The Detective Work: Counting in a Parallel Universe

Counting these shapes directly in the finite field (the pixelated world) is incredibly hard. It's like trying to count every grain of sand on a beach while the tide is coming in.

So, the author uses a clever trick: Time Travel to Characteristic Zero.

The Lift: Imagine lifting these pixelated shapes out of the finite field and into a smooth, infinite world (characteristic zero). In this smooth world, the shapes are easier to study.
The Injection: The author proves a crucial fact: If two shapes are different in the smooth world, they must be different when you drop them back down to the pixelated world. This means if we can count the unique shapes in the smooth world, we have a guaranteed minimum count for the pixelated world.
The Class Group: In the smooth world, the author finds that these shapes correspond to something called a Class Group of a specific ring ( $R_n$ ). Think of the Class Group as a "membership card" system. The size of this group tells us how many different "flavors" of shapes exist.

The Result: A Lower Bound

The author calculates the size of this Class Group for a specific set of integers $n$ (which represent the size of the field extension).

The Finding:
For a "positive density" of integers $n$ (meaning, if you pick a random large number, there's a good chance this rule applies), the size of the isogeny class is at least:

$|I(A, F_{q^n})| \ge q^{n \left( \frac{(g+1)(g-2)}{4} + 1 + o(1) \right)}$

Let's break down that exponent:

$q$ is the number of elements in the base field.
$n$ is the degree of the extension.
$g$ is the dimension of the abelian variety.
The term $\frac{(g+1)(g-2)}{4} + 1$ is the "growth rate."

What does this mean?
It means the family of cousins is huge. It grows exponentially with $n$ . The paper proves this is a lower bound—a guaranteed minimum size. The family is at least this big.

The "Sharpness" Guess

The paper doesn't just stop at the minimum. It makes a conjecture (a very educated guess based on expected behavior). The author suspects that this lower bound is actually the exact size (or very close to it) for most cases.

In other words, the family isn't just "at least this big"; it's likely exactly this big (plus a tiny bit of error that vanishes as numbers get huge). The author writes:

"Conjecture 1.5... we have the following estimate: $|I(A, F_{q^n})| = q^{n \left( \frac{(g+1)(g-2)}{4} + 1 + o(1) \right)}$ ."

What the Paper Rules Out

It rules out the "Supersingular" case for $g=2$ . The author explicitly states that if $g=2$ , the shape is supersingular, and the whole Newton stratum is just one giant isogeny class. That's too simple, so they ignore it.
It rules out the idea that we can easily count these by looking at class groups of all endomorphism rings. The author notes that trying to estimate the upper bound by looking at every possible ring is "out of reach." Instead, they found a specific, clever way to count a subset that gives the right answer.

How Sure Are We?

The Lower Bound: This is proven. The author has mathematically demonstrated that the family is at least this large.
The Exact Estimate (Conjecture): This is suggested. The author believes the lower bound is sharp (meaning the family is exactly that size), but it is currently a conjecture, not a proven theorem.
The Method: The proof relies on counting in a "characteristic zero" world and then mapping it back down. It does not use simulations or guesswork; it uses rigorous algebraic geometry and number theory.

The Takeaway

In the complex garden of Lubin-Tate abelian varieties (where $g > 2$ ), the family of isogenous cousins is massive. The paper proves they are at least as numerous as a specific giant exponential formula predicts, and it strongly suspects that this formula is the exact truth. It's a victory for counting the uncountable by finding a secret door to a smoother world.

Technical Summary: Size of Isogeny Classes of Abelian Varieties of Lubin-Tate Type

Problem Statement
This paper addresses the problem of estimating the size of the isogeny class $I(A, \mathbb{F}_{q^n})$ of a simple principally polarized abelian variety $A$ of dimension $g$ defined over a finite field $\mathbb{F}_q$ . Specifically, the author focuses on the case where $A$ belongs to the Newton stratum of "Lubin-Tate type." This classification requires that the $p$ -divisible group $A[p^\infty]$ is a direct sum of $p$ -divisible groups where the local-local part of each summand (or its dual) has dimension one. Consequently, the Newton polygon of $A$ has slopes $1/g$ and $(g-1)/g$ . The paper assumes $g > 2$ to exclude the supersingular case ( $g=2$ ) and that the endomorphism algebra $K = \mathbb{Q}(\alpha)$ , where $\alpha$ is the Weil $q$ -number of $A$ , is a CM field such that $K \otimes \mathbb{Q}_p$ splits into a direct sum of two unramified extensions of degree $g$ .

The central question is to determine the asymptotic growth of $|I(A, \mathbb{F}_{q^n})|$ as $n$ varies. While heuristics based on the "almost product" structure of Newton strata (central leaves and isogeny leaves) suggest a specific order of magnitude, rigorous lower bounds for non-ordinary, non-almost-ordinary cases have been difficult to establish using previous classification methods.

Methodology
The author employs a strategy distinct from previous work on ordinary and almost-ordinary abelian varieties, which relied on classifying varieties via canonical CM lifts to characteristic zero. Instead, this paper utilizes a "count in characteristic zero, reduce mod $p$ " approach:

Injectivity of Reduction: The paper first establishes that the reduction map from the isogeny class of a lift $\tilde{A}$ over an unramified extension of $\mathbb{Q}_p$ (specifically $L' = W(k)[1/p]$ ) to the isogeny class of its reduction $A$ over $\mathbb{F}_q$ is injective, provided the lifts share the same CM type. This relies on Grothendieck-Messing theory and the classification of finite flat group schemes (Fontaine's Honda systems) to show that endomorphism orders are preserved under reduction for unramified lifts.
Local Conditions on Endomorphism Rings: The author defines a specific order $R_n \subset K$ characterized by local conditions:
- $R_n$ contains $\mathbb{Z}[\alpha^n, (q/\alpha)^n]$ .
- $R_n$ is maximal at $p$ (specifically $R_n \otimes \mathbb{Z}_p \cong \mathcal{O}_{K'} \oplus \mathcal{O}_{K'}$ ).
- At primes $l \neq p$ , $R_n$ is as small as possible, specifically $R_n \otimes \mathbb{Z}_l \cong \mathbb{Z}_l[\alpha^n]$ .
  It is shown that abelian varieties with this specific endomorphism ring $R_n$ descend to $\mathbb{F}_{q^n}$ .
Counting via Class Groups: The problem of counting isomorphism classes of abelian varieties with endomorphism ring $R_n$ is reduced to counting the ideal class group $\text{Cl}(R_n)$ . An injection is constructed from $\text{Cl}(R_n)$ to the set of isogeny classes over $L'$ .
Asymptotic Estimation: The size of the class group $|\text{Cl}(R_n)|$ is estimated using the analytic class number formula. This involves calculating the discriminant of the order $R_n$ and the index $[\mathbb{Z}[\alpha^n] : R_n]$ . The author uses equidistribution arguments (Weyl's criterion) to show that for a set of $n$ with positive natural density, the trigonometric terms in the discriminant formula are bounded away from zero, allowing for a precise asymptotic estimate.
Polarization: Finally, the count is restricted to principally polarized abelian varieties. This is achieved by analyzing the kernel of the norm map from the class group of $R_n$ to the narrow class group of its maximal totally real suborder. The existence of principally polarized varieties in the kernel is established using trace duals and properties of the Weil number.

Key Contributions and Results

Lower Bound Theorem: The paper proves a lower bound for the size of the isogeny class for a set of positive integers $n$ with positive natural density:
$|I(A, \mathbb{F}_{q^n})| \ge q^{n \left( \frac{(g+1)(g-2)}{4} + 1 + o(1) \right)}$
Here, the term $1$ accounts for the dimension of the isogeny leaf, and $\frac{(g+1)(g-2)}{2}$ corresponds to the dimension of the central leaf (divided by 2 in the exponent due to the nature of the count).
Sharpness Conjecture: Based on the expected size of isogeny classes derived from the dimensions of the Newton stratum and the central leaf, the author conjectures that this lower bound is sharp, meaning the size is asymptotically equal to the bound.
Methodological Shift: The paper demonstrates that for Lubin-Tate type varieties, one can bypass the difficulty of finding canonical CM lifts by counting in characteristic zero and relying on the injectivity of the reduction map for varieties with specific endomorphism orders. This contrasts with the classification theorems used for ordinary cases.

Significance and Claims
The author claims that this work extends the understanding of isogeny class sizes to the Lubin-Tate case, a specific non-ordinary Newton stratum where previous classification-based methods were insufficient. The result aligns with the heuristic expectations derived from the geometry of the Newton stratum (specifically the dimensions of the central leaf and isogeny leaf).

The paper does not claim to solve the general problem for all Newton strata or to provide an upper bound for the Lubin-Tate case (noting that finding an upper bound via class group parametrization of non-maximal orders remains "out of reach"). Instead, it provides a rigorous lower bound that matches the heuristic prediction, suggesting the heuristic is likely correct for this class of abelian varieties. The work also generalizes the strategy of counting in characteristic zero to other non-ordinary strata, provided one can control the central leaves.