Congruences and the Iwasawa Main Conjecture for modular forms

Fouquet, Olivier

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 23-42

In the early 2000s, Ralph Greenberg asked whether the Iwasawa Main Conjecture could be proven in a Hida family of nearly-ordinary $p$-adic eigencuspforms by propagating it from a known case through congruences. Emerton-Pollack-Weston showed that this is indeed possible when the $\mu$-invariant of such a family is trivial. In this article, we show that this is the case without this assumption, and that in fact such a result holds in general over the universal deformation ring of an irreducible residual modular Galois representation.

Au début des années 2000, Ralph Greenberg a demandé si l’on pouvait démontrer la Conjecture Principale de la théorie d’Iwasawa pour une famille de Hida de formes modulaires paraboliques propres quasi-ordinaires en la propageant par congruences à partir d’un cas connu. Emerton-Pollack-Weston ont montré que c’était effectivement possible lorsque l’invariant $\mu$ de cette famille est trivial. Dans cet article, nous montrons que c’est le cas sans hypothèse supplémentaire, et que ce résultat demeure en fait vrai sur l’anneau de déformation universelle d’une représentation résiduelle modulaire irréductible.

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