## 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)

### MSC:

 11F85 $$p$$-adic theory, local fields 11F80 Galois representations 11S23 Integral representations

Zbl 0747.00038