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


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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