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Nearly ordinary Hecke algebras and Galois representations of several variables. (English) Zbl 0782.11017
Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 115-134 (1989).
[For the entire collection see Zbl 0747.00038.]
The purpose of this paper is to supplement the author’s previous papers on Hecke algebras over totally real fields with a result on the canonical Galois representations into \(GL_ 2\) with coefficients in the total quotient rings of the Hecke algebras.
More specifically the author proves the following two theorems: Let \(S\) be an open compact subgroup of \(GL_ 2(\prod_{{\mathfrak p}} {\mathfrak r}_{{\mathfrak p}})\) containing \(U_ 1(N)\) with \({\mathfrak r}_{{\mathfrak p}}\) the completion at a prime ideal \({\mathfrak p}\) of a totally real field \(F\) of finite degree, and \({\mathcal O}\) the \(p\)-adic integer ring of a finite extension of the closure of the field generated by all the conjugates of \(F\). \(\mathbf{h} (S;{\mathcal O})\) denotes the full Hecke algebra of infinite \(p\)-power level and \(\mathbf{h}^{n,\text{ord}} (S;{\mathcal O})\) is the nearly ordinary part. Then:
Theorem 1. Let \(A\) be an integral domain of characteristic different from 2 and \(\lambda:\mathbf{h}^{n,\text{ord}} (S;{\mathcal O})\to A\) be a continuous \({\mathcal O}\)-algebra homomorphism. Let \({\mathcal Q}\) be the quotient field of \(A\). Then there exists a unique semisimple Galois representation \(\pi: Gal(\mathbb{Q}/F)\to GL_ 2({\mathcal Q})\) such that: (i) \(\pi\) is continuous; (ii) \(\pi\) is unramified outside \(Np\), where \(N\) is the level of \(S\); (iii) For the Frobenius element \(\varphi_{\mathfrak q}\) for each prime \({\mathfrak q}\) outside \(Np\), \[ \text{det}(1- \pi(\varphi_{{\mathfrak q}})X)= 1-\lambda(T({\mathfrak q})) X+\lambda(\langle {\mathfrak q}\rangle){\mathfrak N}_{F/\mathbb{Q}}({\mathfrak q})X^ 2; \] (iv) Let \({\mathfrak p}\) be a prime factor of \(p\) and fix a decomposition group \(D_{{\mathfrak p}}\) in \(Gal(\overline {\mathbb{Q}}/F)\). Then there exist two characters \(\varepsilon\), \(\delta\) of \(D_{{\mathfrak p}}\) with values in \(A\) such that the restriction of \(\pi\) to \(D_{{\mathfrak p}}\) is, up to equivalence, of the following form: \[ \pi(\sigma)=\left( {{\varepsilon(\sigma)} \atop 0} {*\atop {\delta(\sigma)}} \right) \qquad \text{for } \sigma\in D_{{\mathfrak p}}. \] Moreover if \(A\) and \(\lambda\) satisfy additional conditions which are given specifically in the paper, \(\pi\) is absolutely irreducible.
Theorem 2 is the similar statement to Theorem 1 without (iv) by replacing \(\lambda:\mathbf{h}^{n,\text{ord}}(S;{\mathcal O})\to A\) by \(\lambda:\mathbf{h}\to A\). From this one can associate a canonical Galois representation to any \(p\)-adic common eigenform of all Hecke operators.
Reviewer: K.I.Ohta (Tokyo)

11F85 \(p\)-adic theory, local fields
11F80 Galois representations
11S23 Integral representations