The generic fiber of the universal deformation space associated to a tame Galois representation
Let \(E\) be a finite extension of \(\mathbb{Q}\), \(Q\) a finite set of primes in \(E\), \(\overline\rho: G_E\rightarrow \text{ GL}_2(k)\) a representation of the absolute Galois group of \(E\) over the finite field \(k\) of characteristic \(l\), unramified outside \(Q\). The equivalence classes of lifts of \(\overline\rho\) to complete Noetherian local rings with residue field \(k\) are parametrized by a universal deformation ring \(R_Q\). The author shows: if \(\overline\rho\) is tame and irreducible, and if the Leopoldt conjecture holds for the splitting field of \(\text{ ad}_{\overline\rho}\), then the spectrum of \(R_Q\otimes {\mathbb Q}_l\) (viewed \(l\)-adically analytically) is smooth near the trivial lift. This implies that \(R_Q\) can be written as a quotient of \(W(k)T_1,\dots ,T_n\) (power series over the ring of Witt vectors) by \(m\) relations, where \(n\) and \(m\) are prescribed by cohomological data.
scientific article; zbMATH DE number 1182243