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Tamely ramified Hida theory. (English) Zbl 1048.11043

Let \(N\) be a natural number and \(p\) a prime not dividing \(N\). Let \(X_1(N,p)\) denote the modular curve associated with the group \(\Gamma_1(N,p)= \Gamma_1(N) \cap\Gamma_0(p)\), and \(X_1(Np)\) the curve associated with \(\Gamma_1(Np)\). The authors study the deformation theory obtained from the \(p-1\) torsion in the Jacobians of \(X_1(N,p)\) and \(X_1(Np)\). The work extends previous investigations of the second author [Compos. Math. 95, 69–100 (1995; Zbl 0853.11045); and Invent. Math. 121, 225–255 (1995; Zbl 1044.11576)]. In section 2 they study \(J_1(Np)[p-1]\) as a deformation of its part fixed by the group of \(p\)-diamond operators: Hecke structure, Galois action and filtration for the action of the decomposition group are summarized in Theorem 2.8. Section 3 is devoted to questions of breaking the deformation into components. In section 4 they show how the infinitesimal variation of \(U_p\) in the deformation is related to the \(p\)-adic period matrix of the abelian subvariety of the Jacobian (Theorem 4.6).
Several differences between Hida’s theory of “ordinary” \(p\)-adic deformations and “tame” Hida theory considered in this article are indicated on p. 29.

MSC:

11F85 \(p\)-adic theory, local fields
11G18 Arithmetic aspects of modular and Shimura varieties
11R23 Iwasawa theory
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References:

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