Geometry-of-numbers methods over global fields II: Coregular representations

This paper develops geometry-of-numbers methods to count orbits in coregular vector spaces over global fields, applying these techniques to bound average ranks and determine average Selmer group sizes for elliptic curves and Jacobians of hyperelliptic curves over any global field of characteristic not 2, 3, or 5.

The Gist

Imagine you are a cosmic librarian trying to organize an infinite library of mathematical objects called elliptic curves and hyperelliptic curves. These aren't just any curves; they are the workhorses of modern cryptography and number theory. The problem is, there are infinitely many of them, and they come in all shapes and sizes.

To make sense of this chaos, mathematicians need a way to count them and understand their "average" behavior. Do they usually have a lot of solutions? Do they usually have few? This paper is a massive upgrade to the tools used to do this counting, moving from a single, specific library (the rational numbers, $\mathbb{Q}$) to every possible library in the universe of number theory (called "global fields").

Here is a breakdown of what the authors, Manjul Bhargava, Arul Shankar, and Xiaoheng Wang, have achieved, using simple analogies.

1. The Problem: Counting the Uncountable

Think of an elliptic curve as a specific type of equation (like $y^2 = x^3 + Ax + B$). The numbers $A$ and $B$ determine the shape of the curve.

• The Challenge: There are infinitely many pairs of $(A, B)$. You can't just list them all.
• The Solution: You need a "ruler" to measure them. The authors use a concept called Height. Imagine the Height is the "size" of the numbers $A$ and $B$. A curve with small numbers (like $A=1, B=1$) has a low height. A curve with massive numbers has a high height.
• The Goal: Count how many curves exist with a height less than $X$, and then see what happens as $X$ gets infinitely large. This gives us the "average" behavior of all curves.

2. The Old Way vs. The New Way

Previously, mathematicians could only do this counting effectively for the rational numbers (fractions like $1/2, 3/4$). It was like having a map that only worked for one country.

• The Breakthrough: This paper extends the map to every country in the mathematical world. This includes:
  • Number Fields: Extensions of the rational numbers (like adding $\sqrt{2}$ to the mix).
  • Function Fields: Curves defined over finite fields (think of these as "clock arithmetic" on a geometric shape).
• The Method: They developed a new set of "Geometry of Numbers" techniques. Imagine trying to count how many grains of sand are in a giant, shifting dune. The old methods worked for a dune in the desert (rational numbers). The new methods work for dunes on Mars, in the ocean, or floating in space.

3. The Coregular Representations: The "Magic Filter"

The authors use a sophisticated mathematical structure called a Coregular Representation.

• The Analogy: Imagine a giant, multi-dimensional sieve (a filter). You pour all the possible curves through this sieve.
• The Magic: The sieve is designed so that it separates the curves based on their "invariants" (special properties that don't change even if you rotate or stretch the curve).
• The Result: The sieve sorts the curves into "orbits." Two curves are in the same orbit if they are essentially the same shape, just viewed from a different angle or scale. The authors count these orbits.

4. The Main Discoveries (The "What Did They Find?" Section)

Once they could count these orbits over any global field, they applied the results to three big questions:

A. The Average Rank of Elliptic Curves (The "Complexity" Meter)

Every elliptic curve has a "rank," which roughly measures how many independent solutions it has.

• The Prediction: Mathematicians guess the average rank is $0.5$ (meaning half the curves have rank 0, half have rank 1, or something similar).
• The Result: The authors proved that for any global field (not just rational numbers), the average rank is at most 1.05.
• Why it matters: This is a huge improvement over previous bounds. It suggests that, on average, these curves are not infinitely complex; they are relatively simple.

B. The Size of Selmer Groups (The "Solution Count")

The "Selmer group" is a mathematical object that helps predict the rank.

• The Result: They calculated the average size of these groups for elliptic curves and found it matches a specific formula (the sum of the divisors of $n$). This confirms that the "statistics" of these curves are universal, regardless of which number system you are using.

C. Rational Points on Hyperelliptic Curves (The "Loneliness" of Curves)

Hyperelliptic curves are more complex cousins of elliptic curves.

• The Question: Do these curves have any "rational points" (solutions that fit the rules of the number system)?
• The Result: They proved that for high-degree curves, almost all of them have no rational points at all.
• The Analogy: Imagine throwing darts at a giant board. For small boards, you hit the bullseye often. For these massive, high-degree curves, the board is so huge and the target so tiny that if you throw enough darts, you will almost never hit a target. In fact, as the curves get more complex, the chance of finding a solution approaches 0%.

5. The "Sieve" and the "Cusp"

The paper is famous for its technical depth, specifically in how they handle the "tails" of the distribution.

• The Cusp: Imagine a funnel. Most of the sand (curves) falls through the wide part. But a tiny bit gets stuck in the narrow, long neck (the "cusp").
• The Innovation: In previous work, counting the sand in the neck was hard and required case-by-case analysis. The authors created a set of Axioms (a rulebook). If a mathematical structure follows these rules, you automatically know how to count the sand in the neck, no matter which country (global field) you are in. This turned a messy, manual process into a clean, automated machine.

Summary

This paper is a universal toolkit.

1. It takes a method that only worked for one specific type of number system.
2. It upgrades the method to work for all number systems.
3. It uses this to prove that elliptic curves are generally simple (low rank) and that complex curves are generally empty of solutions.

It's like discovering a universal law of gravity that works not just on Earth, but on every planet in the universe, and using it to predict exactly how many apples fall from trees on each one.

Technical Summary

1. Problem Statement

The paper addresses the challenge of extending arithmetic statistics from the rational numbers ($\mathbb{Q}$) to arbitrary global fields (number fields and function fields of curves over finite fields). Specifically, it aims to count integral orbits in coregular representations of reductive groups with bounded invariants (heights).

While the first paper in this series ([13]) successfully generalized counting methods to prehomogeneous vector spaces (where a group acts with an open dense orbit), many important arithmetic problems involve coregular representations. In coregular cases:

• The ring of invariants is a polynomial ring, but the action is not necessarily prehomogeneous.
• There is no unique "Steinitz class" for orbits, meaning a single rational orbit may correspond to multiple integral orbits across different class groups.
• The fundamental domains required for counting are more complex and require careful construction to handle "cusps" (non-compact regions).

The authors apply these generalized methods to three major problems:

1. Elliptic Curves: Bounding average ranks and determining average sizes of $n$-Selmer groups ($n=2,3,4,5$).
2. Hyperelliptic Curves (Monic): Bounding average ranks of Jacobians.
3. Hyperelliptic Curves (General): Analyzing the rarity of rational points and bounding the size of 2-Selmer sets of principal homogeneous spaces.

2. Methodology

The authors develop a unified framework that generalizes the geometry-of-numbers techniques previously used over $\mathbb{Q}$ to arbitrary global fields. The methodology proceeds in four main stages:

A. General Framework and Axioms

The authors define a set of axioms that a pair $(G, V)$ (a reductive group $G$ acting on a vector space $V$) must satisfy to apply their counting machinery. These axioms are:

1. Representation: $G$ is semisimple, $V$ is coregular, and the sum of invariant degrees equals $\dim V$.
2. Local Weights: Existence of weight functions defined by congruence conditions to handle local solubility and stabilizers.
3. Counting at Infinity: Techniques to count lattice points in fundamental domains over the infinite places ($F_\infty$), specifically handling the "cusp" (non-compact) regions.
4. Local Spreading: Existence of local sections to relate invariants to orbits.
5. Uniformity Estimates: Bounds on the error terms when sieving out non-generic orbits (e.g., those with reducible discriminants).

B. Verification of Axioms over Global Fields

A central technical contribution is proving that if these axioms hold over $\mathbb{Q}$ (which was established in prior literature), they hold over any global field $F$ with minimal extra effort.

• Counting at Infinity: They reduce the verification of the "cutting off the cusp" condition to purely combinatorial conditions on the characters of the maximal split torus of $G$. They prove these conditions are independent of the base field $F$ (provided the representation is base-changed from $\mathbb{Z}$).
• Uniformity Estimates: They generalize the geometric sieve, reduction sieve, and embedding sieve techniques. They show that the error terms for counting lattice points in expanding regions (Davenport's lemma) and sieving conditions extend naturally to function fields and number fields.

C. Handling Class Groups

Unlike the prehomogeneous case where orbits map uniquely to a class group element, coregular orbits may intersect multiple lattices $V_\beta$ associated with the class group $cl(G)$.

• The authors construct a family of lattices $V_\beta$ and subgroups $G_\beta$ for each $\beta \in cl(G)$.
• They introduce a weight function to account for the fact that a single rational orbit might be represented in multiple $V_\beta$, ensuring the final count is not overestimated.

D. Mass Formulas

Once the number of orbits is counted, the authors compute the asymptotic constants using a Tamagawa number formula. This involves computing local volumes (masses) at finite and infinite places, which cancel out via product formulas for global fields and abelian varieties, leaving the Tamagawa number of $G$ as the leading constant.

3. Key Contributions

1. Generalization of Coregular Counting: The paper successfully extends the geometry-of-numbers method from prehomogeneous to coregular representations over arbitrary global fields.
2. Axiomatic Reduction: It establishes that the difficult combinatorial inputs required for counting (specifically regarding the cusp) are purely algebraic properties of the representation, independent of the global field.
3. Uniformity Estimates: It provides the necessary uniformity estimates for sieving over global fields, a critical step for obtaining sharp bounds on Selmer groups.
4. New Arithmetic Statistics: It derives the first general bounds for average ranks and Selmer group sizes for elliptic and hyperelliptic curves over function fields and number fields (beyond $\mathbb{Q}$).

4. Main Results

Theorem 1: Average Rank of Elliptic Curves

For any global field $F$ of characteristic not 2, 3, or 5, when elliptic curves are ordered by height, their average rank is at most 1.05.

• Significance: This significantly improves previous bounds (e.g., $3/2$ for number fields, $4/3$ for function fields) and approaches the conjectured value of $1/2$.

Theorem 2: Average Rank of Hyperelliptic Jacobians

For monic hyperelliptic curves of degree $m$ over a global field $F$ (char $\neq 2$):

• If $m$ is odd, the average rank of the Jacobian is $\le 3/2$.
• If $m$ is even, the average rank is $\le 5/2$.

Theorem 3: Rarity of Rational Points

For locally soluble hyperelliptic curves of genus $n$:

• A positive proportion have no points over any odd-degree extension of $F$.
• As $n \to \infty$, the proportion of curves with no rational points approaches 100%.

Theorem 4: Average Size of $n$-Selmer Groups (Elliptic Curves)

For $n \in \{2, 3, 4, 5\}$, the average size of the $n$-Selmer group of an elliptic curve over $F$ is exactly $\sigma(n)$ (the sum of divisors of $n$).

• This confirms the Cohen–Lenstra heuristics for Selmer groups in this general setting.

Theorem 5: Average Size of 2-Selmer Groups (Hyperelliptic)

For monic hyperelliptic curves of degree $m$:

• If $m$ is odd, the average size of the 2-Selmer group of the Jacobian is $\le 3$.
• If $m$ is even, the average size is $\le 6$.

Theorem 7: Average Size of 2-Selmer Sets (General Hyperelliptic)

For locally soluble general hyperelliptic curves of genus $n$, the average size of the 2-Selmer set of the principal homogeneous space $J_1$ is bounded above by 2.

5. Significance

• Unification of Arithmetic Statistics: The paper unifies the study of arithmetic statistics across number fields and function fields, showing that the underlying geometric and combinatorial structures are field-independent.
• Resolution of Open Problems: It resolves long-standing questions regarding the average rank of elliptic curves over function fields and general number fields, providing the strongest bounds to date.
• Methodological Advancement: By formalizing the "axioms" required for counting, the paper provides a blueprint for applying these techniques to other coregular representations and arithmetic problems in the future.
• Implications for BSD: The bounds on Selmer group sizes provide crucial evidence for the Birch and Swinnerton-Dyer conjecture and the distribution of ranks in families of curves, suggesting that high-rank curves are rare.

In summary, this work represents a major leap in the application of the geometry of numbers to global fields, moving from specific cases over $\mathbb{Q}$ to a robust, general theory applicable to all global fields.