Polarized superspecial abelian varieties over $\mathbb{F}_p$ via hermitian lattices

This paper determines the nonemptiness and classifies the genera of isomorphism classes of polarized superspecial abelian varieties over $\mathbb{F}_p$ with specific Frobenius and polarization kernel conditions by reducing the problem to the arithmetic study of hermitian lattices.

The Gist

Imagine you are an architect trying to build a very specific type of city. This city has strict rules:

1. It must be built on a specific type of soil (a prime number field, $F_p$).
2. The buildings must be "superspecial," meaning they are essentially perfect copies of a few basic, super-strong structures (products of supersingular elliptic curves).
3. The city must have a special "polarization," which is like a unique energy grid connecting all the buildings in a specific way.
4. Most importantly, the city's "Frobenius endomorphism" (a mathematical way of describing how the city transforms or rotates) must act like a square root of negative $p$ ($\sqrt{-p}$).

The paper by Lin, Xue, and Yu is essentially a master blueprint and a census for these cities. They answer three big questions:

• Existence: Can we build such a city at all, or are the rules impossible to satisfy?
• Classification: If we can build them, how many different "types" (genera) of these cities exist?
• The Method: How do we count them without getting lost in the infinite complexity of geometry?

Here is the breakdown using simple analogies.

1. The Problem: Geometry is Hard, Algebra is Easy

The authors are studying "Abelian Varieties." Think of these as complex, multi-dimensional donuts (tori) that exist in the world of number theory. Counting how many distinct versions of these donuts exist is notoriously difficult, like trying to count every unique snowflake in a blizzard.

The Magic Trick (The Lattice Translation):
The authors use a clever translation tool. They realize that instead of studying the complex "donuts" directly, they can translate the problem into studying Hermitian Lattices.

• The Analogy: Imagine the "donut" is a complex 3D sculpture. It's hard to measure. But the authors found that every sculpture has a hidden "shadow" cast on a 2D grid (a lattice).
• If you can count and classify the shadows (the lattices), you automatically know everything about the sculptures (the abelian varieties).
• The paper says: "Let's stop looking at the sculptures and start counting the shadows."

2. The Main Discovery: When Can We Build the City?

The first major result (Theorem 1.1) answers the question: "Is it possible to build this city?"

The answer depends on two things: the "soil" (the prime number $p$) and the "size" of the city (the dimension $n$).

• The Rule: You can build the city unless you have a very specific "bad luck" combination:
  • The soil is a prime number that leaves a remainder of 7 when divided by 8 ($p \equiv 7 \pmod 8$).
  • AND the city size is a number that leaves a remainder of 2 when divided by 4 ($n \equiv 2 \pmod 4$).

The Metaphor:
Imagine you are trying to bake a cake.

• If you use "Flour A" (most primes), you can bake the cake of any size.
• If you use "Flour B" (primes like 3, 7, 11...), you can bake the cake only if the cake size is a multiple of 4.
• If you use "Flour C" (primes like 7, 23...), you can bake the cake only if the size is a multiple of 4.
• The Exception: If you use "Flour C" (specifically $p \equiv 7 \pmod 8$) and try to bake a cake of size 2, 6, 10... (numbers like $2 \pmod 4$), the batter simply won't hold together. The city cannot exist.

3. The Census: How Many Types of Cities?

Once they know the city can exist, they ask: "How many different neighborhoods (genera) are there?"

In math, two cities are in the same "genus" if they look identical when you zoom in very close (locally) at every prime number, even if they look different from a distance.

The authors found that the number of neighborhoods depends on the "soil" and the "size":

• Case 1 (Most soils): There are usually 1 or 2 types of neighborhoods.
• Case 2 (Specific soils like $p \equiv 3 \pmod 4$): The number of neighborhoods explodes!
  • If the city is small, there might be $n/2$ types.
  • If the city is large, there could be $n$ or even $3n/2$ types.

The Metaphor:
Think of the "genus" as the architectural style.

• In some countries (mathematical settings), there are only two styles: "Modern" and "Classic."
• In other countries, the rules are so flexible that if you build a city with 10 buildings, you might find 15 different valid architectural styles. The authors figured out the exact formula for how many styles exist based on the size of the city.

4. The Secret Weapon: The "Bass Order" Decomposition

The hardest part of the paper is the last section (Section 5), where they develop a new way to break down these "shadows" (lattices).

The Analogy:
Imagine you have a complex Lego structure made of different colored bricks. You want to take it apart to count the pieces.

• Usually, you can just pull it apart into individual bricks.
• But sometimes, the bricks are glued together in weird ways (non-maximal orders).
• The authors discovered a universal disassembly tool (Theorem 1.5). They proved that no matter how weirdly the bricks are glued, you can always break the structure down into a stack of simpler, "free" blocks.
• This allows them to count the complex structures by just counting the simple blocks.

Summary

This paper is a mathematical census for a very specific, high-tech type of geometric object.

1. Translation: They turned a hard geometry problem into an easier algebra problem (counting lattice shadows).
2. Existence: They found the exact "recipe" for when these objects can exist (it depends on the prime number and the dimension).
3. Counting: They provided a formula to count exactly how many distinct families (genera) of these objects exist.
4. Innovation: They invented a new way to break down complex algebraic structures into simple pieces, which helps solve this problem and likely many others in the future.

It's like finding a master key that unlocks the door to counting complex shapes by turning them into simple grid patterns.

Technical Summary

Here is a detailed technical summary of the paper "Polarized Superspecial Abelian Varieties over $\mathbb{F}_p$ via Hermitian Lattices" by Yucui Lin, Jiangwei Xue, and Chia-Fu Yu.

1. Problem Statement

The paper investigates the set of isomorphism classes of polarized superspecial abelian varieties $(A, \lambda)$ of a fixed dimension $n$ over the finite field $\mathbb{F}_p$. Specifically, it focuses on a specific subset defined by two conditions:

1. Frobenius Condition: The Frobenius endomorphism $\pi_A$ satisfies $\pi_A = \sqrt{-p}$ (implying $\pi_A^2 = -p$).
2. Kernel Condition: The kernel of the polarization $\lambda$ coincides with the kernel of the Frobenius, i.e., $\ker \lambda = \ker \pi_A = A[F]$, where $F$ is the relative Frobenius morphism.

This set, denoted as $\Sigma_n(\sqrt{-p})$, is crucial for understanding the geometry of the supersingular locus and generalizing Deuring's $2T-H$ Theorem (relating type numbers to the number of fixed points under Galois action).

Key Questions Addressed:

• Existence: Under what conditions on $p$ and $n$ is the set $\Sigma_n(\sqrt{-p})$ non-empty?
• Classification: How can these varieties be classified? Specifically, how many "genera" (isogeny classes with isomorphic $\ell$-divisible groups) exist within this set?
• Galois Action: Does there exist a model defined over $\mathbb{F}_p$ (a fixed point under the Galois group) for these varieties?

2. Methodology

The authors employ a lattice-theoretic approach to translate geometric problems about abelian varieties into arithmetic problems about hermitian lattices.

• Lattice Description: Building on the work of Jordan, Ibukiyama, Karemaker, and Yu, the paper utilizes an equivalence of categories between:
  • The category of polarized abelian varieties over $\mathbb{F}_p$ isogenous to a power of a supersingular elliptic curve $E$ (with $\pi_E = \sqrt{-p}$).
  • The category of positive-definite hermitian lattices over the order $R = \mathbb{Z}[\sqrt{-p}]$.
• Translation of Conditions:
  • The condition $\ker \lambda = A[F]$ translates to the lattice condition $\sqrt{-p} L^\vee = L$, where $L^\vee$ is the dual lattice with respect to the hermitian form. Such lattices are called $\sqrt{-p}$-modular.
  • The distinction between varieties with $\text{End}(A) \supseteq \mathcal{O}_K$ (maximal order) and those with $\text{End}(A) = \mathbb{Z}[\sqrt{-p}]$ (non-maximal order) corresponds to studying lattices over $\mathcal{O}_K$ versus lattices over the suborder $\mathbb{Z}[\sqrt{-p}]$.
• Arithmetic Analysis: The problem is reduced to classifying $\sqrt{-p}$-modular hermitian lattices over local and global fields. The authors use:
  • Local-Global Principles: Analyzing lattices at each prime $\ell$ (especially $\ell=2$ and $\ell=p$).
  • Bass Orders: Utilizing the structural theory of lattices over commutative local Bass orders to handle the non-maximal order case ($p \equiv 3 \pmod 4$).
  • Orthogonal Decomposition: Proving a new decomposition theorem for unimodular lattices over Bass orders to classify them.

3. Key Contributions and Results

A. Existence Theorems (Theorems 1.1 & 1.2)

The paper provides a complete characterization of when the set $\Sigma_n(\sqrt{-p})$ is non-empty. Let $n$ be a positive even integer. The set is non-empty if and only if one of the following holds:

1. $p \not\equiv 3 \pmod 4$;
2. $p \equiv 3 \pmod 4$ and $4 \mid n$;
3. $p \equiv 3 \pmod 8$ and $n \equiv 2 \pmod 4$.

Conversely, the set is empty if and only if $p \equiv 7 \pmod 8$ and $n \equiv 2 \pmod 4$.

• Refinement: The authors distinguish between varieties where the endomorphism ring contains the maximal order $\mathcal{O}_K$ and those where it does not. They prove that the existence of the latter depends on the specific congruence of $p$ modulo 8 and the divisibility of $n$.

B. Classification of Genera (Theorem 1.3)

The authors classify the number of genera (isomorphism classes of $\ell$-divisible groups) in these sets. The number of genera depends on the congruence of $p$ and $n$:

• Case $p \not\equiv 3 \pmod 4$: There is 1 genus if $4 \nmid n$, and 2 genera if$4 \mid n$.
• Case $p \equiv 3 \pmod 4$:
  • If $4 \mid n$: 1 genus for the maximal order case;$n$genera (if$p \equiv 3 \pmod 8$) or$3n/2$genera (if$p \equiv 7 \pmod 8$) for the non-maximal order case.
  • If $n \equiv 2 \pmod 4$: The maximal order case is empty; the non-maximal case has $n/2$ genera.

C. Arithmetic Results on Hermitian Lattices (Theorems 1.4 & 1.5)

A significant portion of the paper is dedicated to the intrinsic arithmetic of hermitian lattices, independent of the geometric application:

• $\sqrt{-p}$-modular Lattices: They determine necessary and sufficient conditions for the existence of $\sqrt{-p}$-modular lattices over $\mathcal{O}_K$ and $\mathbb{Z}[\sqrt{-p}]$.
• Uniform Classification: They provide a uniform proof for the classification of self-dual hermitian lattices over $\mathbb{Z}_2[\sqrt{-p}]$, unifying previous separate treatments for $p \equiv 3 \pmod 8$ and $p \equiv 7 \pmod 8$.
• Orthogonal Decomposition Theorem (Theorem 1.5): They prove that any self-dual hermitian lattice $M$ over a local Bass order $R$ admits an orthogonal decomposition $M = M_1 \perp \dots \perp M_m$, where each $M_i$ is a free lattice over a specific overorder $R_i$. This generalizes classical results (which assumed projective lattices) to general lattices over Bass orders.

4. Significance

• Resolution of Open Problems: The paper resolves the existence question for non-principal superspecial abelian varieties with Frobenius $\sqrt{-p}$ over $\mathbb{F}_p$, a problem that was previously open (specifically regarding the existence of a model over $\mathbb{F}_p$ for the non-principal genus).
• Generalization of Deuring's Theorem: It extends the classical Deuring $2T-H$ relation and Ibukiyama-Katsura's work to the non-principal genus, providing explicit formulas for the number of fixed points.
• Methodological Advancement: The paper successfully bridges the gap between the geometry of moduli spaces of abelian varieties and the arithmetic of hermitian lattices. The developed tools, particularly the orthogonal decomposition theorem for Bass orders, are of independent interest in algebraic number theory and the theory of quadratic forms.
• Explicit Formulas: The results allow for the computation of explicit mass formulas and size estimates for these sets, which are essential for understanding the distribution of superspecial points in moduli spaces.

In summary, this work provides a comprehensive arithmetic classification of a specific class of superspecial abelian varieties by reducing the geometric problem to the classification of hermitian lattices, yielding precise existence criteria and genus counts based on the arithmetic properties of the prime $p$ and the dimension $n$.