By Joseph H. Silverman
In The mathematics of Elliptic Curves, the writer awarded the fundamental idea culminating in primary worldwide effects, the Mordell-Weil theorem at the finite iteration of the gang of rational issues and Siegel's theorem at the finiteness of the set of necessary issues. This ebook keeps the learn of elliptic curves via proposing six vital, yet a little bit extra really expert subject matters: I. Elliptic and modular capabilities for the whole modular crew. II. Elliptic curves with advanced multiplication. III. Elliptic surfaces and specialization theorems. IV. Néron types, Kodaira-N ron type of exact fibres, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's conception of q-curves over p-adic fields. VI. Néron's concept of canonical neighborhood peak capabilities.
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Additional info for Advanced Topics in the Arithmetic of Elliptic Curves
9]. Remark. The simple characters θ, θ0 are endo-equivalent. 2). The group H 1 ∩ N− · J 0 ∩ Q+ admits a unique representation μ+ which is trivial on H 1 ∩ N− and J 0 ∩ N+ , and which restricts to μ0 ⊗ μ0 ⊗ · · · ⊗ μ0 on J 0 ∩ M . Lemma 2. (1) The representation μ ˜+ of J 1 ∩ N− · J 0 ∩ Q+ , induced by ˜ + |J 1 ∼ μ+ , is irreducible, and satisﬁes μ = η. ˜+ . (2) There is a unique representation μ of J 0 extending μ (3) The representation μ lies in H0 (θ). Proof. Assertion (1) is straightforward.
Surely y intertwines κ+ , hence also κ to be a homomorphism relative to the group (J 1 ∩ N− · P × J 0 ∩ Q+ ) ∩ (J 1 ∩ N− · P × J 0 ∩ Q+ )y . So, φ is a homomorphism relative to the group generated by y J 0 ∩ J 0 ∪ (J 1 ∩ N− · P × J 0 ∩ Q+ ) ∩ (J 1 ∩ N− · P × J 0 ∩ Q+ )y . y This surely contains (J 0 ∩ J 0 )P × = J ∩ J y . We have shown that y intertwines κ, as required. The proof of the proposition yields rather more. 2) κ0 −→ κ, using the same notation as in the proof. If T /F is the maximal tamely ramiﬁed sub-extension of P/F , then H(θ0 ) is a principal homogeneous space over X1 (T ) and H(θ) is a principal homogeneous space over X1 (T )/X0 (T )s .
The group Δ acts on E as Gal(E/E0 ). It ﬁxes the endo-class ΘE , so it acts on A01 (E; ΘE ). Let A01 (E; ΘE )Δ-reg be the subset of Δ-regular elements. 6), applied with base ﬁeld K, we get a canonical bijection indE/K : A01 (E; ΘE ) → A01 (K; ΘK ). 3 Corollary 2, this is a Δ-map. 1) indE/F = indK/F ◦ indE/K . 8 Proposition to obtain: Theorem. Let Θ ∈ E(F ) have tame parameter ﬁeld E0 /F . Let E/E0 be unramiﬁed of degree m, and set Δ = Gal(E/E0 ). Let ΘE0 be a totally wild E0 /F lift of Θ, let ΘE be the unique E/E0 -lift of ΘE0 .