Elliptic Modular Forms and Their Applications - Up To

8 D. ZagierAx 2 + Bxy + Cy 2 with A, B, C ∈ Z and B 2 − 4AC = D. Such a form isdefinite (i.e., Q(x, y) ≠0for non-zero (x, y) ∈ R 2 ) and hence has a fixed sign,which we take to be positive. (This is equivalent to A>0.) We also assumethat Q is primitive, i.e., that gcd(A, B, C) = 1.DenotebyQ D the set ofthese forms. The group Γ 1 (or indeed Γ 1 )actsonQ D by Q ↦→ Q ◦ γ, where(Q ◦ γ)(x, y) =Q(ax + by, cx + dy) for γ = ± ( abcd)∈ Γ1 . We claim that thenumber of equivalence classes under this action is finite. This number, calledthe class number of D and denoted h(D), also has an interpretation as thenumber of ideal classes (either for the ring of integers or, if D is a non-trivialsquare multiple of some other discriminant, for a non-maximal order) in theimaginary quadratic field Q( √ D), so this claim is a special case – historicallythe first one, treated in detail in Gauss’s Disquisitiones Arithmeticae –ofthe general theorem that the number of ideal classes in any number field isfinite. To prove it, we observe that we can associate to any Q ∈ Q D theunique root z Q =(−B + √ √ √D)/2A of Q(z, 1) = 0 in the upper half-plane (hereD =+i |D| by definition and A>0 by assumption). One checks easilythat z Q◦γ = γ −1 (z Q ) for any γ ∈ Γ 1 ,soeachΓ 1 -equivalence class of formsQ ∈ Q D has a unique representative belonging to the setQ redD = { [A, B, C] ∈ Q D |−A