On arithmetic invariants of binary cubic and binary trilinear forms

On Canonical Binary Trilinear Forms

Quaternary binary trilinear forms

Dear friends, In 1997 or 1998 I began to write a paper [20], trying to extend the known description of binary forms of norm unity, in the real or complex cases, to the case of quaternions, but I never finished it. In 2002, I resumed my work but then I was unable to understand what I had done, so I made a fresh start with somewhat different assumptions. Now I have realized that this was a gross ...

Reduction of Binary Cubic and Quartic Forms

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over Q. Remarks are given c...

Invariants of Polynomials and Binary Forms

We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.

Cubic Residues and Binary Quadratic Forms

Let p > 3 be a prime, u, v, d ∈ Z, gcd(u, v) = 1, p u2 − dv2 and (−3d p ) = 1, where ( p ) is the Legendre symbol. In the paper we mainly determine the value of u−v √ d u+v √ d (p−( p3 ))/3 (mod p) by expressing p in terms of appropriate binary quadratic forms. As applications, for p ≡ 1 (mod 3) we obtain a general criterion for m(p−1)/3 (mod p) and a criterion for εd to be a cubic residue of p...