Torsion points on $\rm{GL}_2$-type abelian varieties

Inspired by Katz's work, this paper investigates the converse of the known property regarding rational torsion injection into reductions for $\rm GL_2$-type abelian varieties and proposes a conjectural list of possible torsion orders for modular abelian varieties over $\mathbb Q$ with dimension up to 5.

The Gist

Imagine you are trying to figure out how many people are in a secret club (the torsion points) inside a massive, complex building (an abelian variety). This building is defined over a "number field," which is just a fancy way of saying it follows specific mathematical rules based on fractions and whole numbers.

The problem is: The club members are hidden deep inside the building. You can't just walk in and count them. However, you can look at the building's "shadows" or "reflections" projected onto the ground at different locations (these are the reductions modulo primes).

The Old Rule: The Shadow Tells the Truth

Mathematicians have long known a simple rule: If you look at the shadow of the building at a specific spot, the number of people you see in that shadow is always a multiple of the actual number of secret club members inside.

• Analogy: If the real club has 6 members, the shadow might show 6, 12, 18, or 24 people, but it will never show 7 or 5.
• The Logic: Therefore, if you look at shadows from many different spots and find the number of people that is common to all of them (the Greatest Common Divisor), that number must be divisible by the real club size.

The Big Question: Does the Shadow Reveal the Whole Truth?

The authors of this paper asked a deeper question, inspired by a famous mathematician named Nicholas Katz:
"If the shadows at almost every single spot all show a number divisible by $X$, does that guarantee there is a version of the building where the secret club actually has $X$ members?"

For simple buildings (like elliptic curves, which are 1-dimensional), the answer is "Yes."
For very complex, multi-dimensional buildings, the answer used to be "No, sometimes the shadows trick you."

The Breakthrough: The "GL2-Type" Buildings

This paper focuses on a specific, important class of buildings called GL2-type abelian varieties. Think of these as buildings that have a special, symmetrical internal structure (like a specific type of crystal lattice) that makes them behave more predictably than random buildings.

The authors proved a powerful theorem:
For these special GL2-type buildings, the shadows do tell the whole truth.

If the shadows at almost every location suggest the club size is a multiple of a certain number, then there definitely exists a version of this building (one that is mathematically "isogenous," or structurally equivalent) where the secret club actually has that many members.

How They Did It (The Detective Work)

To prove this, the authors didn't just count people; they looked at the "blueprints" of the building's symmetry (called Galois representations).

• The Metaphor: Imagine the building has a security system that changes its code based on who is looking at it. The authors showed that if the security code (the shadow) is "open" (divisible by a number) almost everywhere, then the actual security system must have a "master key" (a rational torsion point) that opens the door.

The Result: A "Wanted" List for Club Sizes

Using this new rule, the authors wrote a computer program (using a system called Magma) to scan thousands of these special buildings. They calculated what the "maximum possible club size" could be for buildings of dimensions 2, 3, 4, and 5.

They produced a "Conjectural List" of possible club sizes.

• Example: For a 2-dimensional building, the club size could be 1, 2, 3, ... up to 56, but it cannot be 23 or 25 (based on their data).
• Why it matters: This helps mathematicians fill in the gaps in databases like the LMFDB (a massive library of mathematical objects). They found, for instance, a specific building that should have a club of 28 members, a fact that was missing from previous records.

In Summary

1. The Problem: We can see "shadows" of hidden mathematical points, but we weren't sure if the shadows always revealed the true size of the hidden group.
2. The Discovery: For a special, important family of mathematical shapes (GL2-type), the shadows always reveal the truth.
3. The Application: The authors used this to create a "menu" of all possible sizes for these hidden groups, helping to organize and expand our knowledge of these complex mathematical structures.

It's like finally realizing that if you check the footprints of a hidden animal in the mud at 99% of the locations, you can be 100% sure of exactly how heavy the animal is, provided the animal belongs to a specific, well-behaved species.

Technical Summary

Here is a detailed technical summary of the paper "Torsion Points on GL2-Type Abelian Varieties" by Jessica Alessandrì and Nirvana Coppola.

1. Problem Statement and Context

The paper addresses a fundamental question in arithmetic geometry regarding the relationship between the rational torsion of an abelian variety and the number of points on its reduction modulo primes.

• The Local-Global Principle: It is a standard fact that for an abelian variety $A$ defined over a number field $K$, the group of rational torsion points, $\text{Tors}(A)(K)$, injects into the group of points of the reduction $\tilde{A}(\mathbb{F}_p)$ for any prime $p$ of good reduction. Consequently, the order of the torsion group divides the greatest common divisor (GCD) of the sizes of these reduction groups, $N(p) = |\tilde{A}(\mathbb{F}_p)|$.
• Katz's Question (1981): Nicholas Katz posed the converse: If $N(p) \equiv 0 \pmod m$ for a set of primes of density one, does there exist an abelian variety $A'$ isogenous to $A$ such that $|\text{Tors}(A')(K)| \equiv 0 \pmod m$?
  • Katz proved this is true for elliptic curves.
  • He proved a weaker version for abelian surfaces (relating prime divisors rather than exact orders).
  • He provided counterexamples for higher-dimensional abelian varieties in the general case.
• The Gap: While bounds on torsion exist for specific cases (e.g., elliptic curves over $\mathbb{Q}$ via Mazur, or surfaces with potential quaternionic multiplication), no uniform bound or complete classification exists for general higher-dimensional abelian varieties. Specifically, the behavior of GL2-type abelian varieties in arbitrary dimensions was not fully resolved regarding Katz's converse.

2. Methodology

The authors employ a strategy combining Galois representation theory and local-global analysis to reformulate and solve Katz's problem for a specific, broad class of varieties.

• GL2-Type Abelian Varieties: The study focuses on abelian varieties $A$ of dimension $g$ over a number field $K$ such that there is an embedding of a number field $F$ (of degree $g$) into $\text{End}_K(A) \otimes \mathbb{Q}$. This class includes modular abelian varieties associated with newforms.
• Galois Representations: The authors utilize the $\ell$-adic Galois representations $\rho_\ell: G_K \to \text{Aut}(T_\ell(A))$. For GL2-type varieties, these decompose into a compatible system of 2-dimensional representations $\rho_\lambda: G_K \to \text{GL}_2(F_\lambda)$, where $\lambda$ is a prime of $F$ above $\ell$.
• Reformulation via Lattices: The condition $N(p) \equiv 0 \pmod m$ is translated into a condition on the characteristic polynomial of Frobenius: $P_p^\lambda(1) \equiv 0 \pmod{\lambda^n}$. The problem is then recast as finding a $G_K$-stable lattice $L' \subseteq L$ such that the quotient $L/L'$ is a trivial $G_K$-module of a specific order.
• Application of Katz-Lang Theorem: The authors adapt a theorem by Katz and Lang regarding 2-dimensional representations over local fields. They prove that if $\det(1-g) \equiv 0 \pmod{\pi^n}$ for all $g$ in a compact subgroup, then the representation can be conjugated into a specific subgroup of matrices, guaranteeing the existence of a stable lattice with a trivial quotient.

3. Key Contributions and Theoretical Results

The paper establishes a positive answer to Katz's "weaker" question (regarding prime divisors) for GL2-type abelian varieties in all dimensions.

Main Theorem (Theorem 3.2 & Corollary 3.3)

Let $A$ be a GL2-type abelian variety over a number field $K$. Let $\Sigma$ be the set of primes of good reduction where the absolute ramification index is small ($e_p < p-1$).

1. Prime Divisor Equivalence: The set of rational primes dividing the order of the torsion group of some variety $A'$ isogenous to $A$ is exactly the same as the set of primes dividing $\gcd_{p \in \Sigma} N(p)$.
2. Valuation Bounds: For any rational prime $\ell$ of good reduction:
   $$v_\ell\left( \sup_{A' \sim_K A} |\text{Tors}(A')(K)| \right) \leq v_\ell\left( \gcd_{p \in \Sigma} N(p) \right)$$
3. Equality Condition: If $\ell$ is totally inert in the coefficient field $F$, the inequality becomes an equality:
   $$v_\ell\left( \sup_{A' \sim_K A} |\text{Tors}(A')(K)| \right) = v_\ell\left( \gcd_{p \in \Sigma} N(p) \right)$$

This result implies that for GL2-type varieties, the "local" data (point counts modulo primes) completely determines the "global" torsion structure up to isogeny, at least regarding the prime factors and their valuations under specific inertial conditions.

4. Computational Implementation and Conjectures

The authors applied their theoretical results to modular abelian varieties over $\mathbb{Q}$ (which correspond to weight 2 newforms via Serre's conjectures).

• Algorithm: They implemented a Magma script to:
  1. Iterate over newforms of weight 2 and level up to 500.
  2. Compute the GCD of $P_p^\lambda(1)$ for primes $p$ up to a bound (heuristically the Sturm bound).
  3. Predict the maximal possible torsion order for varieties in the isogeny class.
• Findings:
  • For dimensions $g=2, 3, 4$, the predicted torsion orders matched the GCD of point counts exactly for levels up to 500.
  • For dimension $g=5$, a discrepancy was observed for a specific newform (LMFDB label 399.2.a.f), where the GCD of point counts (72) was strictly larger than the predicted torsion order (36). This suggests the theoretical bound might not always be sharp in higher dimensions without further constraints.
• Conjectural Lists: Based on the computational data, the authors propose Conjecture 4.4 and Conjecture 4.5, which list the possible torsion orders and the set of possible prime divisors for simple GL2-type abelian varieties of dimensions $g=2$ to $5$.
  • Example: For $g=2$, the possible torsion orders include 1 through 24, plus 28, 44, 56. The possible prime divisors are $\{2, 3, 5, 7, 11, 13, 19\}$.

5. Significance

• Generalization of Mazur's Theorem: While Mazur's theorem classifies torsion for elliptic curves ($g=1$), this work extends the classification framework to higher dimensions ($g \leq 5$) for a significant class of varieties (GL2-type).
• Resolution of Katz's Converse: It provides a definitive positive answer to Katz's local-global divisibility question for GL2-type varieties, a class that was previously known to have counterexamples in the general case.
• Data Enhancement: The paper provides new data points for the LMFDB (L-functions and Modular Forms Database). For instance, it identifies an abelian surface with torsion order 28 associated with level 39, which was previously missing from standard databases.
• Methodological Bridge: The paper successfully bridges abstract Galois representation theory with concrete computational number theory, offering a practical algorithm to predict torsion structures for modular abelian varieties.

In summary, the paper proves that for GL2-type abelian varieties, the local point counts modulo primes are sufficient to determine the prime factors of the global torsion group, and it leverages this to generate the most comprehensive conjectural lists of torsion orders for modular abelian varieties up to dimension 5.