TY - JOUR

T1 - On the number of integral binary n-ic forms having bounded Julia invariant

AU - Bhargava, Manjul

AU - Yang, Andrew

N1 - Funding Information:
We thank John Cremona, Peter Sarnak, Arul Shankar, and Michael Stoll for helpful conversations. This work was done in part while the authors were at MSRI during the special semester on Arithmetic Statistics. The first author was supported by a Simons Investigator Grant and NSF grant DMS‐1001828.
Publisher Copyright:
© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

PY - 2022

Y1 - 2022

N2 - In 1848, Hermite introduced a reduction theory for binary forms of degree (Formula presented.) 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 (Formula presented.) -invariant of binary (Formula presented.) -ic forms defined over (Formula presented.), which is now known as the Julia invariant. In this paper, for each (Formula presented.) and (Formula presented.) with (Formula presented.), we determine the asymptotic behavior of the number of (Formula presented.) -equivalence classes of binary (Formula presented.) -ic forms, with (Formula presented.) pairs of complex roots, having bounded Julia invariant. Specializing to (Formula presented.) 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.

AB - In 1848, Hermite introduced a reduction theory for binary forms of degree (Formula presented.) 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 (Formula presented.) -invariant of binary (Formula presented.) -ic forms defined over (Formula presented.), which is now known as the Julia invariant. In this paper, for each (Formula presented.) and (Formula presented.) with (Formula presented.), we determine the asymptotic behavior of the number of (Formula presented.) -equivalence classes of binary (Formula presented.) -ic forms, with (Formula presented.) pairs of complex roots, having bounded Julia invariant. Specializing to (Formula presented.) 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.

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U2 - 10.1112/blms.12625

DO - 10.1112/blms.12625

M3 - Article

AN - SCOPUS:85130506364

SN - 0024-6093

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

ER -