[Paper Review] On the number of integral binary $n$-ic forms having bounded Julia invariant

Manjul Bhargava, Andrew Yang|arXiv (Cornell University)|Dec 27, 2013

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper establishes the asymptotic number of SL₂(ℤ)-equivalence classes of integral binary n-ic forms with k pairs of complex roots and bounded Julia invariant, proving that the count grows as cₙ,ₖX^{(n+1)/2} + O(X^{(n+1)/2 − 1/n}) for n + k ≥ 3. The result generalizes Gauss’s and Davenport’s classical theorems on binary quadratic and cubic forms, respectively, and provides a uniform framework for counting integral forms via Julia’s invariant, a fundamental SL₂(ℝ)-invariant introduced in 1917.

ABSTRACT

In 1848, Hermite introduced a reduction theory for binary forms of degree $n$ which was developed more fully in the seminal 1917 treatise of Julia. This canonical method of reduction made use of a new, fundamental, but irrational $\mathrm{SL}_2$-invariant of binary $n$-ic forms defined over $\mathbb{R}$, which is now known as the Julia invariant. In this paper, for each $n$ and $k$ with $n+k\geq 3$, we determine the asymptotic behavior of the number of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of binary $n$-ic forms, with $k$ pairs of complex roots, having bounded Julia invariant. Specializing to $(n,k)=(2,1)$ and $(3,0)$, respectively, recovers the asymptotic results of Gauss and Davenport on positive definite binary quadratic forms and positive discriminant binary cubic forms, respectively.

Motivation & Objective

• To determine the asymptotic growth rate of the number of SL₂(ℤ)-equivalence classes of integral irreducible binary n-ic forms with k pairs of complex roots and bounded Julia invariant.
• To refine prior Oε(X^{(n+1)/2 + ε}) bounds to an exact asymptotic with a power-saving error term.
• To unify and extend classical results on binary quadratic and cubic forms by treating them as special cases of a general invariant-based counting framework.
• To establish that the Julia invariant serves as a natural and effective parameter for classifying integral binary n-ic forms under SL₂(ℤ)-equivalence.

Proposed method

• The authors use Julia’s SL₂(ℝ)-invariant θ(f), defined via a canonical positive-definite quadratic covariant Q of f, to define a bounded region in the space of binary n-ic forms.
• They analyze the volume of a fundamental domain R_X(L) associated with the Julia invariant θ(f) ≤ X, using integration over a symmetric space and group-theoretic averaging.
• The main counting function Nₙ,ₖ(X) is related to the volume of this region via a lattice point counting argument, with error terms controlled by integrals over the stabilizer and cusp regions.
• The proof adapts techniques from the geometry of numbers and ergodic theory, particularly using the structure of the group SL₂(ℝ) and its action on the space of forms.
• The authors handle reducible forms via a separate error term estimate, showing their contribution is O(X^{(n+1)/2 − 1/n}).
• For congruence conditions, they derive a weighted version of the asymptotic by incorporating local densities μₚ(S) at each prime p, using scaling and volume normalization.

Experimental results

Research questions

• RQ1 How many SL₂(ℤ)-equivalence classes of integral binary n-ic forms with k pairs of complex roots have Julia invariant at most X?
• RQ2 Can the prior Oε(X^{(n+1)/2 + ε}) bound on the number of such forms be refined to an exact asymptotic with a power-saving error term?
• RQ3 Does Julia’s invariant provide a uniform and effective parameter for classifying integral binary n-ic forms across all degrees n and signatures k?
• RQ4 What is the precise asymptotic constant cₙ,ₖ in the growth of Nₙ,ₖ(X), and how does it relate to known constants in the quadratic and cubic cases?

Key findings

• The number of SL₂(ℤ)-equivalence classes of integral binary n-ic forms with k pairs of complex roots and Julia invariant ≤ X is asymptotically cₙ,ₖX^{(n+1)/2} + O(X^{(n+1)/2 − 1/n}) for n + k ≥ 3.
• The asymptotic constant cₙ,ₖ is given by the volume of a fundamental domain in the space of forms, normalized by the stabilizer volume, and is positive for all valid n and k.
• For (n,k) = (2,1), the constant c₂,₁ is π/36, recovering Gauss’s class number formula for positive definite binary quadratic forms.
• For (n,k) = (3,0), the constant c₃,₀ is π²/36, recovering Davenport’s asymptotic for positive discriminant binary cubic forms.
• The error term O(X^{(n+1)/2 − 1/n}) is power-saving and sharp, matching the second-order terms found by Shintani in the classical cases.
• The result extends to sets defined by finitely many congruence conditions, with the asymptotic modified by a product of local densities μₚ(S) over all primes p.