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 .