By Haruzo Hida

This e-book presents a accomplished account of the idea of moduli areas of elliptic curves (over integer earrings) and its software to modular kinds. the development of Galois representations, which play a basic position in Wiles' evidence of the Shimura - Taniyama conjecture, is given. furthermore, the booklet provides an summary of the facts of various modularity result of two-dimensional Galois representations (including that of Wiles), in addition to many of the author's new ends up in that path. during this new moment variation, a close description of Barsotti - Tate teams (including formal Lie teams) is extra to bankruptcy 1. As an program, a down-to-earth description of formal deformation thought of elliptic curves is included on the finish of bankruptcy 2 (in order to make the facts of regularity of the moduli of elliptic curve extra conceptual), and in bankruptcy four, although constrained to boring situations, newly integrated are Ribet's theorem of complete photograph of modular p-adic Galois illustration and its generalization to 'big' lambda-adic Galois representations lower than gentle assumptions (a new results of the author). although a number of the extraordinary advancements defined above is out of the scope of this introductory booklet, the writer provides a style of latest examine within the sector of quantity conception on the very finish of the e-book (giving a great account of modularity idea of abelian Q-varieties and Q-curves).

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Since ϕ normalizes the subgroup Gd,e = g we have ϕ−1 ◦g◦ϕ = g k for some k ∈ N and so g ◦ ϕ = ϕ ◦ g k . Hence g(ϕ(Cx )) = ϕ(g k (Cx )) = ϕ(Cx ) and, similarly, g(ϕ(Cy )) = ϕ(Cy ). This yields (25). e. ϕ(¯0) = ¯0. Now the last assertion follows easily. 10. If e2 ≡ 1 mod d then + − Nd,e = Nd,e , Nd,e , (27) while for e2 ≡ 1 mod d, (28) + + Nd,e = Nd,e , Nd,e = Nd,e ,τ . Proof. For ϕ ∈ Nd,e we let C = ϕ−1 (Cy ). 9 (see (25)) the cyclic group Gd,e stabilizes C and ¯0 ∈ C. 8 there is + − , Nd,e which sends C to one of the coordinate an automorphism ψ ∈ Nd,e −1 axes Cx , Cy .

10. (a) Let C = {f (y) = 0}, where f ∈ C[y] is a polynomial of degree ≥ 2 with simple roots. If K ⊆ C denotes the set of these roots, then Stab(C) = T1,0 · U + · Stab(K) , where U + ⊆ Aut(A2 ) is as in (6), T1,0 = {λ ∈ T | λ : (x, y) → (αx, y), α ∈ C× } , and the stabilizer Stab(K) ⊆ Aut(A1 ) → Aut(A2 ) acts naturally on the symbol y. (b) If C of type (II) is the coordinate cross {xy = 0} in A2 then Stab(C) = N (T) is the normalizer of the maximal torus T in the group GL(2, C). (c) If C of type (IV) is a union of r aﬃne lines through the origin, where r ≥ 3, then Stab(C) ⊆ GL(2, C) is a ﬁnite extension of the group T1,1 = C× · id of scalar matrices.

Hence this ﬁber cannot carry a curve with Euler characteristic 1. This leads to a contradiction, because d > 1 by our assumption. Thus the curve C 1 is smooth. It follows that every ﬁber of p is isomorphic to A1 . Hence there is an automorphism δ ∈ Aut(A2 ) sending the curves C i to the lines y = κi with distinct ki , where κ1 = 1. Moreover, we may suppose that δ(¯0) = ¯0. 8. 5). Hence the elements g = ρ(g) ∈ T and ρ(g ) ∈ T are conjugated in GL(2, C) and so either ρ(g ) = ρ(g) = g or ρ(g ) = τ gτ .