A local model for the trianguline variety and applications

Christophe Breuil; Eugen Hellmann; Benjamin Schraen

doi:10.1007/s10240-019-00111-y

Article

A local model for the trianguline variety and applications

Christophe Breuil ¹; Eugen Hellmann ¹; Benjamin Schraen ¹

¹

Publications Mathématiques de l'IHÉS, Volume 130 (2019), pp. 299-412

Abstract

We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck’s simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties.

Received: 2017-02-22
Accepted: 2019-07-01
Online First: 2019-08-22
Published online: 2019-12-07

MR Zbl

DOI: 10.1007/s10240-019-00111-y

Author's affiliations:

Christophe Breuil ¹; Eugen Hellmann ¹; Benjamin Schraen ¹

¹

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     author = {Christophe Breuil and Eugen Hellmann and Benjamin Schraen},
     title = {A local model for the trianguline variety and applications},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {299--412},
     year = {2019},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {130},
     doi = {10.1007/s10240-019-00111-y},
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     zbl = {1454.14120},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-019-00111-y/}
}

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Christophe Breuil; Eugen Hellmann; Benjamin Schraen. A local model for the trianguline variety and applications. Publications Mathématiques de l'IHÉS, Volume 130 (2019), pp. 299-412. doi: 10.1007/s10240-019-00111-y

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