By Robin Hartshorne, C. Musili

ISBN-10: 3540051848

ISBN-13: 9783540051848

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C ˇ/ŒK:Q we first observe the following. If p 2 P, then max¹1, j˛ C ˇjp Ä max¹1, max¹j˛jp , jˇjp ºº Ä max¹1, j˛jp max¹1, jˇjp . ˇ/jº. Note that, if in the last inequality we let run through EK and we multiply the resulting relations, then the factor 2ŒK:Q will appear in the right-hand side. 16). ˛/. We use the identity max¹1, x 1 º D x 1 max¹1, xº, valid for every real x > 0. 10). ˛ 1 /, as required. 3 Heights: Absolute and logarithmic Heights over Q. Let us specialise our previous discussion to the important case of rational numbers.

That theorem requires that the ai ’s be algebraic integers. In our proposition, these coefficients are 3 One can easily see that there always exists such a model D. b2 =12/, which appear in Silverman’s theorem, by log jj and logC jj j, logC jb2 =12j, respectively. 07, where 1 and j1 are, respectively, the discriminant and the j -invariant of the model D. 43) holds and j1 D j , the j -invariant of E. 07. Ä/ C h. log 2 C h. log 2 C h. 44)). 42). Remark. 3, the Weierstrass model D must fulfil certain conditions, but otherwise it is arbitrary.

1 there exists a Weierstrass model C : y 2 C a1 xy C a3 y D x 3 C a2 x 2 C a4 x C a6 with coefficients in K. P // 7! P / 2 K and this function is even (x. P /). 35) where h. / denotes the absolute logarithmic height. This limit is independent from the Weierstrass model. x 0 , y 0 / D 0 be another Weierstrass model over K of E. P /C for convenient Ä, 2 K. P // 7! P / 2 K and this function is even (x 0 . P /). 2N P C // 1 . P indication on P . 2 P / . 9]. As already noted, for S. 28). 28), the canonical height used by S.

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