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Galois representations into $$\text{GL}_2(\mathbb Z_p[[X]])$$ attached to ordinary cusp forms. (English) Zbl 0612.10021
Let $$p$$ be a prime $$\geq 5$$, $$N$$ an integer prime to $$p$$, $$\Gamma$$ the topological group $$1+p{\mathbb Z}_ p$$, and $$\Omega$$ a $$p$$-adic completion of an algebraic closure of $${\mathbb Q}_ p$$. The author uses the study of the parabolic cohomology groups of $$\Gamma_ 1(Np^ r)$$ to obtain results about Galois representations attached to cusp forms. Beginning with a study of the ordinary part $$h^ 0(N,{\mathbb Z}_ p)$$ of the universal Hecke algebra (generated by the Hecke operators regarded as endomorphisms of the space of $$p$$-adic cusp forms of level $$N$$) the author proves the following theorem concerning $$q$$-expansions: Let $$\Lambda ={\mathbb Z}_ p[[X]]$$ be the one variable Iwasawa algebra, and fix a non-trivial $$\Lambda$$-algebra homomorphism $$\lambda: h^ 0(N,{\mathbb Z})\to \Lambda$$. The image under $$\lambda$$ of the formal power series $$\sum^{\infty}_{n=1}T(n)q^ n$$ is, when evaluated at $$X=\varepsilon (u)u^ k-1$$ $$(u=1+p)$$, the complex $$q$$-expansion of a common eigenform $$f_{k,\varepsilon}$$ for all $$T(n)$$ in $$S_ k(\Gamma_ 1(p^ r))$$ where $$\varepsilon: \Gamma\to \Omega^*$$ is of finite order, $$\text{ker}\;\varepsilon=1+p^ r {\mathbb Z}_ p$$, and $$k$$ is an integer $$\geq 2$$. Moreover, $$f_{k,\varepsilon}$$ is not the twist by any Dirichlet character of a form of lower level.
The author then shows that one can associate to $$\lambda$$ a unique Galois representation $$\pi(\lambda): \text{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \text{GL}_ 2(\Lambda)$$ such that the reduction of $$\pi(\lambda)$$ is equivalent to the irreducible $$\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$$-representation $$\pi(f_{k,\varepsilon})$$ into $$\text{GL}_ 2(\Omega)$$ associated to $$f_{k,\varepsilon}$$ (by Eichler-Shimura for $$f=2$$, and by Deligne for $$k>2$$).
The paper ends with a study of the connections between the special values of the $$L$$-function associated to the 3-dimensional subrepresentation contained in $$\pi (f_{k,\varepsilon})\otimes {\tilde \pi}(f_{k,\varepsilon})$$ (where $${\tilde \pi}$$ is the contragredient representation of $$\pi$$), and the characteristic power series of a certain Iwasawa module that is associated to $$\lambda$$.
Reviewer: S.Kamienny

##### MSC:
 11F80 Galois representations 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F33 Congruences for modular and $$p$$-adic modular forms 11F11 Holomorphic modular forms of integral weight 11F25 Hecke-Petersson operators, differential operators (one variable) 11R23 Iwasawa theory
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