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On \(p\)-adic analytic families of Galois representations. (Appendix by Nigel Boston). (English) Zbl 0654.12008

Hida has produced continuous Galois representations \(\rho_ p: G_{\mathbb Q}\to \text{GL}_ 2(\mathbb Z_ p[[t]])\) such that the specializations \(t\mapsto (1+p)^{k-1}- 1\) \((k=2,3,...)\) are \(p\)-adic representations \(\rho_ p^{(k)}\) attached by Deligne to cuspidal newforms of weight \(k\).
In this paper the geometry behind Hida’s construction is studied. It involves the tower of Jacobians \(J_ 1(p^ n)\) of the modular curves \(X_ 1(p^ n)\); the contravariant Tate-modules \[ W_ n=\text{Hom}(J_ 1(p^ n)({\overline {\mathbb Q}}),{\mathbb Q}_ p/{\mathbb Z}_ p) \] and their projective limit \(W\). This \(W\) is a Galois module and a module for the Hecke algebra \(T\). For a suitable maximal ideal \({\mathfrak m}\) of \(T\), the completion \(T_{{\mathfrak m}}\) is isomorphic to \({\mathbb Z}_ p[[ t]]\) and \(W_{{\mathfrak m}}=W\otimes T_{{\mathfrak m}}\) is a free \(T_{{\mathfrak m}}\)-module of rank 2. Hence \(W_{{\mathfrak m}}\) induces a Galois representation \(\rho_ p\) as above. The representation \(\rho_ p\) is unramified outside \(p\). The action of the inertia group at \(p\) is studied via \(p\)-adic Hodge theory. For the representation attached in this way to the cusp form \(\Delta\) of weight 12 and level 1 one finds that for many values of \(p\) the image of \(\rho_ p\) contains \(\text{SL}_ 2({\mathbb Z}_ p[[ t]])\). Further \(\rho_ p^{(1)}\) is in general not the Deligne-Serre representation attached to a newform of weight one; \(\rho_ p^{(1)}\) is not semisimple and its \(p\)-adic Hodge structure is not semisimple.

MSC:

11R39 Langlands-Weil conjectures, nonabelian class field theory
11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
11R37 Class field theory
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14G25 Global ground fields in algebraic geometry
14G20 Local ground fields in algebraic geometry
11F80 Galois representations
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