On the number of integral binary n$n$‐ic forms having bounded Julia invariant

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 SL 2 -invariant of binary 𝑛-ic forms defined over ℝ, which is now known as the Julia invariant. In this paper, for each 𝑛 and 𝑘 with 𝑛 + 𝑘 ⩾ 3, we determine the asymptotic behavior of the number of SL 2 (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. M S C ( 2 0 2 0 )11E76, 11R45 (primary) INTRODUCTIONLet 𝑉 𝑛 (ℝ) denote the (𝑛 + 1)-dimensional real vector space of binary 𝑛-ic formshaving coefficients 𝑎 0 , … , 𝑎 𝑛 ∈ ℝ. The group SL 2 (ℝ) acts naturally on 𝑉 𝑛 (ℝ) via linear substitution of variable; namely, an element 𝛾 ∈ SL 2 (ℝ) acts on 𝑓(𝑥, 𝑦) by 𝛾 ⋅ 𝑓(𝑥, 𝑦) = 𝑓((𝑥, 𝑦) ⋅ 𝛾).(2)This action of SL 2 (ℝ) on 𝑉 𝑛 (ℝ) is a left action, that is, (𝛾 1 𝛾 2 ) ⋅ 𝑓 = 𝛾 1 ⋅ (𝛾 2 ⋅ 𝑓).