By Haruzo Hida

ISBN-10: 9810243375

ISBN-13: 9789810243371

This publication presents a entire 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 function in Wiles' evidence of the Shimura-Taniyama conjecture, is given. moreover, the e-book provides an summary of the evidence of numerous modularity result of two-dimensional Galois representations (including that of Wiles), in addition to many of the author's new ends up in that course.

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**Haruzo Hida's Geometric Modular Forms and Elliptic Curves PDF**

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**Sample text**

Define a cohomological correspondence ! uC : ( j T g )∗ (iC1 TC1 )! LC1 −→ T 1 (iC2 TC2 )! 1), and the direct image with compact support is defined by duality). Finally, write −1 NC = [K(2) N : h2 K h2 ∩ Nnr (Af )]. 3 The coefficient uC1 ,C2 in the above matrix is equal to NC uC , C where the sum is taken over the set of C ∈ CP such that Tg (C ) = C1 and T1 (C ) = C2 . 5). 5 of [P2], and it is true as well for the Shimura varieties considered here. 7. 5. , that G is not an orthogonal group). Fix an algebraic closure F of Fp .

UC : ( j T g )∗ (iC1 TC1 )! LC1 −→ T 1 (iC2 TC2 )! 1), and the direct image with compact support is defined by duality). Finally, write −1 NC = [K(2) N : h2 K h2 ∩ Nnr (Af )]. 3 The coefficient uC1 ,C2 in the above matrix is equal to NC uC , C where the sum is taken over the set of C ∈ CP such that Tg (C ) = C1 and T1 (C ) = C2 . 5). 5 of [P2], and it is true as well for the Shimura varieties considered here. 7. 5. , that G is not an orthogonal group). Fix an algebraic closure F of Fp . Let V ∈ Ob RepG .

Then, for every z ∈ (RE/Q Gm,Q ), µ(z) = (z, 1)Ip 0 0 (1, z)Iq . 2 Let p ∈ {1, . . , n}. Define a cocharacter µp : Gm,E −→ GE by µp (z) = (z, 1)Ip 0 (1, z)In−p . 2 PARABOLIC SUBGROUPS Let G be a connected reductive algebraic group over Q. Fix a minimal parabolic subgroup P0 of G. Remember that a parabolic subgroup of G is called standard if it contains P0 . Fix a Levi subgroup M0 of P0 . Then a Levi subgroup M of G will be called standard if M is a Levi subgroup of a standard parabolic subgroup and M ⊃ M0 .

### Geometric Modular Forms and Elliptic Curves by Haruzo Hida

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