Constructing certain special analytic Galois extensions

Abstract

For every prime p ≥ 5 for which a certain condition on the class group Cl(Q(µ p )) is satisfied, we construct a p-adic analytic Galois extension of the infinite cyclotomic extension Q(µ p ∞ ) with some special ramification properties. In greater detail, this extension is unramified at primes above p and tamely ramified above finitely many rational primes and its Galois group over Q(µ p ∞ ) is isomorphic to a finite index subgroup of SL 2 (Z p ) which contains the principal congruence subgroup. For the primes 107, 139, 271 and 379 such extensions were first constructed by Ohtani and Blondeau. The strategy for producing these special extensions at an abundant number of primes is through lifting two-dimensional reducible Galois representations which are diagonal when restricted to p.