Download Algebraic Number Theory: Proceedings of an Instructional by J. W. S. Cassels, A. Frohlich PDF

By J. W. S. Cassels, A. Frohlich

This booklet offers a brisk, thorough therapy of the rules of algebraic quantity thought on which it builds to introduce extra complex issues. all through, the authors emphasize the systematic improvement of innovations for the categorical calculation of the fundamental invariants comparable to jewelry of integers, type teams, and devices, combining at every one degree conception with specific computations.

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However, x(P ) is an integer, and none of 3, 6, 11 or 38 is a perfect cube. e. the only torsion point is O). 7. Let p ≥ 2 be a prime number and let us define a curve Ep : y 2 = x3 + p2 . Since x3 + p2 = 0 does not have any rational roots, Ep (Q) does not contain points of order 2. Let P be a torsion point on Ep (Q). The list of all squares dividing 4A3 + 27B 2 = 27p4 is short, and by the Nagell-Lutz theorem the possible values for y(P ) are: y = ±1, ±p, ±p2 , ±3p, ±3p2 , and ± 3. 8. Thus, the torsion subgroup of Ep (Q) is isomorphic to Z/3Z, for any prime p ≥ 2.

1) We define ∆E , the discriminant of E, by ∆E = −16(4A3 + 27B 2 ). For a definition of the discriminant for more general Weierstrass equations, see for example [Sil86], p. 46. 4), and such that the discriminant of E is an integer. The minimal discriminant of E is the integer ∆E that attains the minimum of the set {|∆E | : E ∈ S}. In other words, the minimal discriminant is the smallest integral discriminant (in absolute value) of an elliptic curve that is isomorphic to E over Q. If E is the model for E with minimal discriminant, we say that E is a minimal model for E.

Examples of each of the possible torsion subgroups over Q. isomorphic to G. See, for example, [Kub76], Table 3, p. 217. For the convenience of the reader, the table in Kubert’s article is reproduced in Appendix E. 4. Let Eb : y 2 + (1 − b)xy − by = x3 − bx2 with b ∈ Q and ∆(b, c) = b5 (b2 − 11b − 1) = 0. Then, the torsion subgroup of Eb (Q) contains a subgroup isomorphic to Z/5Z, and (0, 0) is a point of exact order 5. Conversely, if E : y 2 = x3 + Ax + B is an elliptic curve with torsion subgroup equal to Z/5Z then there is an invertible change of variables that takes E to an equation of the form Eb , for some b ∈ Q.

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