Muhammad Manji (University of Warwick)

The Iwasawa main conjecture was stated by Iwasawa in the 1960s, linking the Riemann Zeta function to certain ideals coming from class field theory, and proved in 1984 by Mazur and Wiles. This work was generalised to the setting of modular forms, predicting that analytic and algebraic constructions of the p-adic L-function of a modular form agree. This was proved by Kato (’04) and Skinner–Urban (’06) for ordinary modular forms. For the non-ordinary case there are some modern approaches which use p-adic Hodge theory and rigid geometry to formulate and prove cases of the conjecture. I will review these cases and discuss my work in the setting of automorphic representations of unitary groups, where a new approach uses the L-analytic regulator map of Schneider—Venjakob. My aim is to state a version of the conjecture which was previously unknown, and discuss what is still needed to prove the conjecture in full.