Reductions of Galois representations and the theta operator

Let [Formula: see text] be a prime, and let cuspidal eigenform of weight at least level coprime to finite slope text]. denote the mod Galois representation associated with cyclotomic character. Under an assumption on text], we prove that there exists such up semisimplification. The proof uses Hida–Coleman families theta operator acting overconvergent forms. structure reductions local representations cusp forms slopes in interval were determined by Deligne, Buzzard Gee for Bhattacharya, Ganguli, Ghate, Rai Rozensztajn. We show these reductions, spite their somewhat complicated behavior, are compatible displayed equation above. Moreover, above allows us predict shape class attached eigenforms larger than Finally, methods this paper allow obtain upper bounds radii certain Coleman families.