Elliptic Curves: Diophantine Analysis by Serge Lang

By Serge Lang

It is feasible to write down forever on elliptic curves. (This isn't a threat.) We deal right here with diophantine difficulties, and we lay the rules, particularly for the speculation of critical issues. We evaluation in short the analytic conception of the Weierstrass functionality, after which care for the mathematics points of the addition formulation, over entire fields and over quantity fields, giving upward thrust to the idea of the peak and its quadraticity. We observe this to fundamental issues, overlaying the inequalities of diophantine approximation either at the multiplicative workforce and at the elliptic curve at once. therefore the booklet splits clearly in components. the 1st half offers with the normal mathematics of the elliptic curve: The transcendental parametrization, the p-adic parametrization, issues of finite order and the gang of rational issues, and the relief of convinced diophantine difficulties by way of the speculation of heights to diophantine inequalities concerning logarithms. the second one half offers with the proofs of chosen inequalities, at the least powerful adequate to procure the finiteness of vital points.

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Ifhe looks at Tate [Ta IJ, he will see that the proofs go through in general. We let as before t = xlY. Let r be a positive real number. We now use the notation: A(r) = set of points P in AK such that P is not integral, and ord t(P) ~ r. 1. 1. Let s = l1Y. Then there is a power series expansion where An is a polynomial of weight n in the a i with coefficients ~0 in Z. Proof We deal with the simple Weierstrass form Suppose inductively we have found such that We want to find sm+ 1 to satisfy the congruence mod t m + 5.

Let r be a positive real number. We now use the notation: A(r) = set of points P in AK such that P is not integral, and ord t(P) ~ r. 1. 1. Let s = l1Y. Then there is a power series expansion where An is a polynomial of weight n in the a i with coefficients ~0 in Z. Proof We deal with the simple Weierstrass form Suppose inductively we have found such that We want to find sm+ 1 to satisfy the congruence mod t m + 5. To make the coefficient of t m + 5 on the left equal to the coefficient of t m + 5 on the right, we see at once that it suffices that Am+ 1 be a polynomial in a, band Sk (with k ~ m) with positive integer coefficients, as desired.

L. ljJ(t ' )2 (J/q)6 is valid for t, t' #- qn for all integers n. Observe that even though rj; occurs in the definition of 1jJ, nevertheless the half power disappears in the formula for the difference x(t) - x(t' ) . In fact, the formula amounts to an identity in power series in q, with coefficients which are rational functions in t, t'. g. as in [L 2], Chapter 4, § 2. The formula then remains valid when the variables take on special values in a field complete under a non- 29 § 8. q-Expansions and Products archimedean absolute value, a situation which is discussed in Chapter HI, § 5, under conditions of convergence, namely iqi < L Next we wish to give an estimate for the difference between the Neron function and the local height, defined by hv(P) = log max {l, Ix(P)I} where v here denotes the ordinary absolute value on the complex numbers.

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