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The Iwasawa Main Conjectures for GL2

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Abstract

We prove the one-, two-, and three-variable Iwasawa-Greenberg Main Conjectures for a large class of modular forms that are ordinary with respect to an odd prime p. The method of proof involves an analysis of an Eisenstein ideal for ordinary Hida families for GU(2,2).

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Notes

  1. In the case of elliptic curves this conjecture was first stated by Mazur and Swinnerton-Dyer [46].

  2. This makes no real difference: the modules of (ordinary) p-adic modular forms defined using either choice of Borel are isomorphic; one passes from one setting to the other by conjugation.

  3. This section was written before [39] was available. We made no effort to reconcile our notation with that in loc. cit.

  4. This follows from Theorem 5.3, which shows that the minimal compactification over F p (over which H is defined) is the same as the base change to F p of the minimal compactification over \(\mathcal{O}_{\mathfrak{p}}\).

  5. Proposition 6.6 is no longer true if \(\mathcal{K} \) is of degree >2 over Q since the rank of the group of units is positive.

  6. Not every ordering may occur as a p-stabilization.

  7. In the statement of this theorem we use our fixed identification \(\iota_{p}'\) of \({\overline{{\mathbf{Q}}}}_{p}\) with C.

  8. Recall that we use a geometric convention for Hodge-Tate weights.

  9. More recent modularity results may not require (dist) f at this point, but the hypothesis still figures into later arguments.

  10. Here and throughout we will often view ψ as a function on the A-points of \(\operatorname{Res}_{\mathcal{K}/{\mathbf{Q}}}{\mathbf{G}_{m}} \), so for a place w of Q, ψ w =⊗ v|w ψ v .

  11. The construction in [54] is a specialization of one of the measures constructed by Hida in [28].

  12. This is mostly a technical point. This choice has no effect on the properties of the Fourier coefficients we are interested in.

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Acknowledgements

Both authors gratefully acknowledge their many fruitful conversations with colleagues, especially Ralph Greenberg, Michael Harris, Haruzo Hida, and Andrew Wiles. It also pleasure to thank Xin Wan and the referees for their careful reading of this paper and their helpful comments. The research of the first named author was supported by grants from the National Science Foundation and by a fellowship from the David and Lucile Packard Foundation. The second named author was supported at different times by the CNRS and by grants from the National Science Foundation and by a fellowship from the Guggenheim Foundation.

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  1. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Christopher Skinner

  2. Institut Mathématiques de Jussieu, 175, rue de Chevaleret, 75013, Paris, France

    Eric Urban

  3. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Eric Urban

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Skinner, C., Urban, E. The Iwasawa Main Conjectures for GL2 . Invent. math. 195, 1–277 (2014). https://doi.org/10.1007/s00222-013-0448-1

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