On Quasi-Reduced Quadratic Forms
On Quasi-Reduced Quadratic Forms作者机构:Eugene Dubois UMR 6138 CNRS Dep. de mathematiques et mecanique BP 5186 U. de Caen Basse-Normandie 14032 Caen France Claude Levesque CICMA et Departement de mathematiques et de statistique U. Laval Quebec P. Qudbec Canada G1K 7P4
出 版 物:《Acta Mathematica Sinica,English Series》 (数学学报(英文版))
年 卷 期:2010年第26卷第8期
页 面:1425-1448页
核心收录:
学科分类:07[理学] 070104[理学-应用数学] 0701[理学-数学]
主 题:Quadratic forms reduced forms equivalence of forms class numbers quadratic fields,continued fractions
摘 要:With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 + bXY + cY2 : a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 + bXY + cY2 = (a, b, c) of discriminant △〉0(and of sign σ(f) equal to the sign of a), the quadratic forms associated with f and defined by {〈a+bu+cu2,b+2cu.c〉},with 1≤σ(f)u≤b/2|c| (whenever they exist), 〈c,-b-2cu,a+bu+cu2〉 with b/2|c|≤σ(f)u≤[w(f)]=[b+√△/2|c|], are all different from one another and build a set I(f) whose cardinality is #I(f)={1+[ω(f)],when(2c)|b,[ω(f)],when (2c)|b. If f and g are two different reduced quadratic forms, we show that I(f) ∩ I(g) = Ф. Our main result is that the set Q△ is given by the disjoint union of all I(f) with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #(Q△) involving sums of partial quotients of certain continued fractions.