Tamely ramified Hida theory. (Q5956589)

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).NEWLINENEWLINE Several differences between Hida's theory of ``ordinary'' \(p\)-adic deformations and ``tame'' Hida theory considered in this article are indicated on p. 29.