By Dieudonne J.

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Z/ vary in a 1-1 way from the smallest root e3 to 1. Now we study the behaviour of the function } 0 . z/ e1 . 21 . 1 C! 21  3 x 7! 21 to e3 . 21 . z/ D 0 if z is a vertex. z/ belong to either the interval Œe2 , e1  (if z is on the right vertical side) or to the interval . R/. R/ are those of the two horizontal sides. 1. Case  > 0. R/. z runs along half-open line segments . 1 2 ................... z/ 0 >0 0 <0 0 . 1 2 . . . . . . . . . . . . 1 2 e1 % C1 & e1 0 % C1 # jump 1 % 0 infinite the vertices.

2 P / . 9]. As already noted, for S. 28). 28), the canonical height used by S. 2N P / . 1. 6 The canonical height O / so that D is also a model of E. 37), respectively, are related by O /. Q/. 3]): The Néron–Tate pairing is bilinear. P O O h. P /. P O / D 0 if and only if P E is a torsion point. P hPi , Pj imi mj . 7]. P1 , : : : , Pr / D . 39) (cf. 38)). 2. 7). 39). Proof. P where m is the column vector with components m1 , : : : , mr . As H is symmetric, a def D 1 < 2 < < r of H and an diagonal matrix ƒ of eigenvalues 0 < 26 Chapter 2 Heights orthogonal matrix Q exist such that H D QT ƒQ.

C/ ! r// to r C ƒ. We have to show that is a group homomorphism. 3” of [45]. C/. zi //. P2 /. This is obviously true if at least one Pi is the zero point, therefore we assume that both P1 , P2 are non-zero points. z2 /. 3 Actually, is a group isomorphism; see the beginning of next section. z2 /. 31]. P1 C P2 /. P2 /. z2 /. z1 /. O/ D ƒ. }. z2 /, 12 } 0 . mod ƒ/. r// with r 2 P . P /. ei , 0/ for some i 2 ¹1, 2, 3º (cf. 1 C! , 2 º and, on the other hand, 2P D O. P /. r/ ¤ 0. r/2 C A/. 2P /. C/ 7 !