By Haruzo Hida

ISBN-10: 019857102X

ISBN-13: 9780198571025

The 1995 paintings of Wiles and Taylor-Wiles unfolded a complete new approach in algebraic quantity thought and, a decade on, the waves as a result of this really vital paintings are nonetheless being felt. This e-book, authored through a number one researcher, describes the amazing purposes which have been stumbled on for this method. within the booklet, the deformation theoretic ideas of Wiles-Taylor are first generalized to Hilbert modular kinds (following Fujiwara's treatment), and a few purposes came upon by means of the writer are then mentioned. With many routines and open questions given, this article is perfect for researchers and graduate scholars coming into this learn quarter.

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**Sample text**

More generally, if R is a normal ﬁnite extension of Λ and if Sel∗F (L ⊗R R∗ ) is R-torsion, is char(Sel∗F (L ⊗R R∗ )) a principal ideal generated by fL ∈ R? If that is the case, what is the function fL in the structure sheaf of Spf(R)? Deformation and adjoint square selmer groups 31 Is it related to a p-adic L-function? ) If L is algebro-geometric in the sense of (q1), we expect to have a padic analytic method of constructing p-adic L-functions Lp (L, s) such that Lp (L, s) = φL (γ s − 1) (γ = 1 + p) for φL (x) ∈ W [[x]] and φL (x) = fL (x) up to units in W [[x]], where W is a discrete valuation ring containing W .

Thus ΩA/B represents the covariant functor M → DerB (A, M ) from the category of continuous A-modules into Z-M OD. Since DerB (A, M ) only depends on the image of B in A under the algebra homomorphism ι : B → A giving the B-algebra structure of A, we have ΩA/B ∼ = ΩA/ι(B) . 3) The construction of ΩA/B is easy. The multiplication a ⊗ b → ab induces a B-algebra homomorphism m : A⊗B A → A taking a ⊗ b to ab. We put I = Ker(m), which is an ideal of A⊗B A. Then we deﬁne ΩA/B = I/I 2 . 48). ) given by φ → φ ◦ d.

By local class ﬁeld theory, we have the Artin reciprocity map Fp× u → [u, Fp ] ∈ Dpab such that N ([u, Fp ]) = NFp /Qp (u)−1 if u is a p-adic unit. 4) stable under the decomposition group Dp and an open subgroup of the inertia group Ip ⊂ Dp acts on each subquotient Fpi V /Fpi+1 V by N i . The Galois module V is called nearly p-ordinary if V satisﬁes the above condition for all the prime factors p|p in F . 4) with unramiﬁed Fp0 V /Fp1 V for p|p. If V is p-ordinary for all prime factors p|p, we simply call V p-ordinary.

### Hilbert Modular Forms and Iwasawa Theory by Haruzo Hida

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