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By J-P. Serre

This publication is split into elements. the 1st one is solely algebraic. Its goal is the category of quadratic varieties over the sector of rational numbers (Hasse-Minkowski theorem). it truly is completed in bankruptcy IV. the 1st 3 chapters comprise a few preliminaries: quadratic reciprocity legislations, p-adic fields, Hilbert symbols. bankruptcy V applies the previous effects to critical quadratic different types of discriminant ± I. those varieties ensue in a number of questions: modular features, differential topology, finite teams. the second one half (Chapters VI and VII) makes use of "analytic" equipment (holomor­ phic functions). bankruptcy VI offers the facts of the "theorem on mathematics progressions" as a result of Dirichlet; this theorem is used at a serious aspect within the first half (Chapter unwell, no. 2.2). bankruptcy VII bargains with modular varieties, and specifically, with theta capabilities. many of the quadratic different types of bankruptcy V reappear right here. the 2 elements correspond to lectures given in 1962 and 1964 to moment yr scholars on the Ecole Normale Superieure. A redaction of those lectures within the kind of duplicated notes, was once made via J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They have been very priceless to me; I expand the following my gratitude to their authors.

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We omit the proof of this lemma. Conversely, let E be an elliptic curve over an arbitrary scheme S. 23. 23. 22 coincides with Definition 1 . 1 1 . If, further, char(K) i2, then these definitions coincide with Definition 1 . 1 . Let E be an elliptic curve over S , and let S' ---+ S b e a morphism of schemes. Then, the base change Es' ---+ S' is an elliptic curve over S' . 1, E defines an elliptic curve over L. This is a special case of the base change. 13 signify the following. 24. Let E be an elliptic curve over Q, and let p be a prime number.

Let E be an elliptic curve defined over a field K with char(K) # 2. We say that all the 2-torsion points are K-rational if for any extension L of K, the set of points of order 2 in E(L) is contained in E(K) . 4. Let K be a field with char(K) # 2, and let E be an elliptic curve over K . The fallowing conditions are equivalent. (i) All 2-torsion points of E are K-rational. 6) y 2 = x(x - n) (x - m ) . PROOF . Suppose E is defined by the equation y2 = f (x) . Let L be an extension of K. We prove that a point P = ( s, t) # 0 in E(L) is of order 2 if and only if f ( s ) = t = 0.

We show that a = 1 or 2 . This can be done as follows. Let p be an odd prime. By the change of coordinates x = ux' - v, y = wy', the equation becomes P ROO F O F y'2 = :23 (x' �) (x' v : n ) (x' v : m ) . 8 ) , then v/u, (v+n) /u, and (v+n) /u are all elements in Z ( p ) , and they do not become all equal modulo p. Thus, we may assume v = 0, and we see that u is invertible modulo p. Since au3 /w 2 is also invertible, a is also invertible modulo p. Suppose p = 2. Similarly to the case of an odd prime, two of v/u, (v+n)/u and (v+m) /u are in Z ( 2 ) , and 4 times the other is invertible in F 2 .

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