On the Reduction and Classification of Binary Cubics which Have a Negative Discriminant

Crible et 3-rang des corps quadratiques

“…-Voir [7] et [17]. D Aux constantes près, on a les mêmes majorations que dans le lemme 3.4 (voir [7], lemme 1]) : LEMME 3.6.…”

“…Par contre, dans le cas réel, les automorphismes du Hessien forment un groupe monogène infini, donc la réduction d'Hermite est inadaptée (pour tout ce qui a trait aux classes de formes quadratiques, automorphismes, réduction, nous renvoyons le lecteur au précis de Buell [2]). La réduction des formes de discriminant négatif (due à Mathews et Berwick, voir [7] et [17]) aboutit alors à un domaine fondamental différent.…”

Error estimates for the Davenport-Heilbronn theorems

“…Unfortunately he was not counting orders but classes of binary cubic forms modulo Γ := SL(2, Z), not modulo GL(2, Z) as we need them. Using reduction theory for binary cubics originating in Hermite [15] and Mathews-Berwick [16], he obtained a bijection between classes (modulo Γ) of forms of discriminant ∆, 0 < ±∆ < X, and integer points in semi-algebraic sets C , with a > 0, is associated to the Γ-class of ax 3 + bx 2 y + cxy 2 + dy 3 . Davenport then proceeded to count these integer points.…”

Reduction of Binary Cubic and Quartic Forms

“…Finally, we compare our bounds with the results given in [ , which is best possible since x 3 + x 2 − 2x − 1 has = 49. Now the constant appearing here is 1/ √ 7 = 0.3780, which is slightly smaller than the constant 2/3 √ 3 = 0.3849 which appears in our bound (13). However, Mordell's theorem does not state that the equivalent cubic which minimizes the leading coefficient is actually reduced, so that one cannot deduce, as we did above, that the seminvariant P is simultaneously bounded.…”