here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

146 Chapter 5. Complex multiplication and elliptic curves5.2.3 Binary quadratic formsThe theory of fractional ideals of orders in quadratic number fields is closely linked to that ofbinary quadratic forms. The latter one is especially convenient for computations.Definition 5.2.31 (Binary quadratic form). A binary quadratic form is an element of Z[X, Y ]of the formf = aX 2 + bXY + cY 2 .Its discriminant is ∆ = b 2 − 4ac.Definition 5.2.32 (Primitive form). A binary quadratic form f = aX 2 + bXY + cY 2 is said tobe primitive if gcd(a, b, c) = 1.Definition 5.2.33 (Definite form). A binary quadratic form is said to be positive (respectivelynegative) definite if ∆ < 0 and a > 0 (respectively a < 0).We denote by B(∆) the set of positive definite forms of discriminant ∆, and by b(∆) thesubset of primitive forms.Definition 5.2.34 (Reduced form). A primitive positive definite form f = aX 2 + bXY + cY 2 issaid to be reduced if |b| ≤ a ≤ c, and b ≥ 0 if either |b| = a or a = c.It is possible to define an action of SL 2 (Z) on binary quadratic forms, just as for points of thePoincaré upper halfplane( )p qDefinition 5.2.35. Let σ = ∈ SLr s 2 (Z) and f = aX 2 + bXY + cY 2 a binary quadraticform. We define σf asσf = a(pX + qY ) 2 + b(pX + qY )(rX + sY ) + c(rX + sY ) 2 .Two binary quadratic forms f and g are said to be (properly) equivalent if there exists σ ∈ SL 2 (Z)such that σf = g.Moreover, this action respects the discriminant of the form, its positiveness and the value ofgcd(a, b, c). We also have canonical representatives for each class.Theorem 5.2.36 ([61, Theorem 2.8], [32, Theorem 5.7.7]). Every primitive positive definite formis equivalent to a unique reduced from.We can now define the equivalents of class groups for binary quadratic forms.Definition 5.2.37. Let ∆ be a negative integer such that ∆ ≡ 0, 1 (mod 4). The form classgroup is defined asPic(∆) = b(∆)/ SL 2 (Z) ,and we denote its cardinality, the class number, by h(∆). The form class semigroup is defined asCl(∆) ∗ = B(∆)/ SL 2 (Z) ,and we denote its cardinality, the Kronecker class number, by H(∆).The associated values are then connected by a similar equality.Proposition 5.2.38 ([232]). Let ∆ be a negative integer such that ∆ ≡ 0, 1 (mod 4) andd a positive integer such that d 2 |∆ and ∆/d 2 ≡ 0, 1 (mod 4). Then there is a one-to-onecorrespondence between the sets {f ∈ B(∆) | gcd(a, b, c) = d} / SL 2 (Z) and b(∆/d 2 )/ SL 2 (Z).