## 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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### Online Encyclopedia of Integer Sequences:

Primes p such that tau(p) is congruent to 1 (mod p), where tau is the Ramanujan tau function.

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