Vincent Pilloni, of Université Paris-Saclay, presents a research seminar talk given as part of the Joint IAS/Princeton University Arithmetic Geometry seminar at the Institute for Advanced Study. He addresses the obstruction to lifting the Frobenius map on certain Shimura varieties, a technical problem in arithmetic geometry that connects to the classical theory of companion forms for modular forms, work originally associated with Gross, Coleman-Voloch, and Faltings-Jordan. Pilloni extends these ideas beyond the classical one-dimensional setting into higher-dimensional Shimura varieties, showing how geometric obstructions to Frobenius lifts control which companion forms can exist. The talk assumes graduate-level familiarity with modular forms, Galois representations, and the geometry of Shimura varieties, and proceeds through the obstruction theory before turning to applications. It is a specialist research lecture rather than an introductory survey, aimed at working number theorists and arithmetic geometers following current developments in the field.