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On ordinary $$\lambda$$-adic representations associated to modular forms. (English) Zbl 0664.10013
Let F be a totally real field, $${\mathfrak O}_ F$$ its ring of integers, $${\mathfrak c}$$ an ideal of $${\mathfrak O}_ F$$, and $$\psi$$ a character of ($${\mathfrak O}/{\mathfrak c})^*$$. Let f be a primitive Hilbert modular form of weight k, level $${\mathfrak c}$$, and character $$\psi$$. Suppose that $$T({\mathfrak p})f=c({\mathfrak p},f)f$$ for each Hecke operator $$T({\mathfrak p})$$ associated to an ideal $${\mathfrak p}$$ of $${\mathfrak O}$$. The modular form f is said to be ordinary at the prime $$\lambda$$ of $${\mathfrak O}_ f$$ if for each prime $${\mathfrak p}$$ dividing the norm of $$\lambda$$, the equation $$x^ 2-c({\mathfrak p},f)x+\psi ({\mathfrak p})N{\mathfrak p}^{k-1}=0$$ has at least one root which is a unit mod $$\lambda$$. (Here $${\mathfrak O}_ f$$ is the integer ring of the field generated by the c($${\mathfrak p},f).)$$
Using Hida’s theory of $$\Lambda$$-adic newforms the author shows that if f is a primitive modular form which is ordinary at $$\lambda$$ then there exists a continuous $$\lambda$$-adic representation $$\rho_{\lambda}: Gal(\bar F/F)\to GL_ 2({\mathfrak O}_{\lambda})$$ unramified outside $${\mathfrak c}\cdot (N\lambda)$$ and such that for all primes $${\mathfrak q}\nmid {\mathfrak c}\cdot (N\lambda),\quad trace \rho_{\lambda}(Frob_{{\mathfrak q}})=c({\mathfrak q},f)$$ and $$\det \rho_{\lambda}(Frob_{{\mathfrak q})=\psi ({\mathfrak q}})N{\mathfrak q}^{k-1}$$ (where $$Frob_{{\mathfrak q}}$$ is a $${\mathfrak q}$$-Frobenius automorphism in Gal$$(\bar F/F)$$).
Moreover, in weight one the image of $$\rho_{\lambda}$$ is finite and lifts to a complex 2-dimensional representation. In addition, the restriction of $$\rho_{\lambda}$$ to a $${\mathfrak p}$$-decomposition group is described explicitly. The existence of such representations has long been conjectured (whether or not f is ordinary). The conjecture was already known in various cases including when $$[F:{\mathbb{Q}}]$$ is odd.
Reviewer: S.Kamienny

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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