Lifting of nearly extraordinary Galois representations
Let \(\bar{\rho} : G_{\mathbb{Q}} = \text{Gal}(\bar{\mathbb{Q}} / \mathbb{Q}) \rightarrow \mathrm{GL}_2(\mathbb{F}_p)\) be a continuous, two-dimensional mod \(p\) Galois representation which is odd and unramified outside \(p\). Suppose that the image of \(\bar{\rho}\) contains \(\mathrm{SL}_2(\mathbb{F}_p)\). Suppose also that \(p \geq 5\) and that the restriction \(\bar{\rho} _{| G_{\mathbb{Q} _p}}\) of \(\bar{\rho}\) to the decomposition group at \(p\) is nearly extraordinary, i.e. isomorphic to the sum of two characters. Under these conditions, the main theorem of this article states that there exists a lift \(\rho : G_{\mathbb{Q}} \rightarrow \mathrm{GL}_2(\mathbb{Z}_p)\) of \(\bar{\rho}\) and a finite set of primes \(T\) containing \(p\) such that \(\rho\) is \(T\)-ramified and \(\rho _{|G_{\mathbb{Q}_p}}\) is isomorphic to the sum of two characters. Actually, the author is able to prove a more general version of the above theorem for an arbitrary number field \(K\). The method of the proof follows the work of Taylor and Ramakrishna. It consists of studying a deformation functor associated to the property ''nearly extraordinary at \(p\)'' and denoted \(C_p ^{\text{neo}}\). One studies a pair \((C_p ^{\text{neo}}, L_p ^{\text{neo}})\), where \(L_p ^{\text{neo}}\) is the tangent space of \(C_p ^{\text{neo}}\). In order to show the existence of a deformation of \(\bar{\rho}\) as above, one needs to prove that the pair \((C_p ^{\text{neo}}, L_p ^{\text{neo}})\) is admissible in the language of Taylor (who formalizes the work of Ramakrishna) and moreover that a certain Selmer group is zero. The Selmer group comes from \(L_p ^{\text{neo}}\) and the author proves that it is zero after allowing additional ramification at some subset \(T\). This is done by explicit computations in the Galois cohomology. As an application of his results, the author proves the existence of a Galois extension \(M\) over \(\mathbb{Q} (\mu _{p^{\infty}})\) which is unramified at places dividing \(p\) and such that its Galois group is isomorphic to \(\mathrm{SL}_2(\mathbb{Z}_p)\).
scientific article; zbMATH DE number 6154107