By Joseph H. Silverman
In The mathematics of Elliptic Curves, the writer provided the fundamental idea culminating in basic worldwide effects, the Mordell-Weil theorem at the finite new release of the crowd of rational issues and Siegel's theorem at the finiteness of the set of critical issues. This booklet keeps the learn of elliptic curves by means of offering six very important, yet just a little extra really good issues: I. Elliptic and modular features for the whole modular staff. II. Elliptic curves with advanced multiplication. III. Elliptic surfaces and specialization theorems. IV. Néron types, Kodaira-N ron class of distinctive fibres, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's thought of q-curves over p-adic fields. VI. Néron's thought of canonical neighborhood top capabilities.
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Additional info for Advanced Topics in the Arithmetic of Elliptic Curves
1). Since the canonical class of an abelian surface is trivial, KX is in the image of Z/6 in Pic X, and hence it is a generator of that image. Thus the types of both torsors are the same (up to sign). Hence the pair (Y , the action of µ6 ) can be identiﬁed with the pair (C1 × C2 , the action of µ6 ). Let A be the Albanese variety of Y . This is an abelian surface deﬁned over k. Let s be the k-endomorphism of A given by s = σ∈µ6 σ. Let A1 (respectively A2 ) be the connected component of 0 in ker(s) (respectively in Carmen Laura Basile and Alexei Skorobogatov 35 Aµ6 ).
Vol. XIX, Amer. Math. I. (1967), pp. 111–116  F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970) 169–170  H. P. F. Swinnerton-Dyer, On a problem of Littlewood concerning Riccati’s equation, Proc. Cambridge Phil. Soc. 65 (1969) 651–662  H. P. F. Swinnerton-Dyer, The birationality of cubic surfaces over a given ﬁeld, Michigan Math. J. 17 (1970) 289–295 Bibliography 25  H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms, Acta Arith.
3). Call this point Q. The inverse image of Q in D deﬁnes a class ρ ∈ H 1 (k, µ6 ) = k ∗ /k ∗6 . Consider the twisted torsor E ρ × Dρ → X. Now Dρ has a k-point over Q. But the action of µ6 on E preserves the origin, hence the twisted curve E ρ has a k-point. Therefore, we obtain a k-point on E ρ × Dρ , and hence on X. Note that for the bielliptic surfaces of Corollary 2 the quotient of Br X by the image of Br k is inﬁnite, but in the proof we only used the Brauer–Manin conditions given by the elements of the conjecturally ﬁnite group X(J ).
Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman