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We give a proof of the slope classicality theorem in classical and higher Coleman theory for modular curves of arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of GL 2 ( ℚ p ) . The U p operator becomes a double coset, and unitarity yields slope vanishing.
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Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology
Abstract We construct a $$(\mathfrak {gl}_2, B(\mathbb {Q}_p))$$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $$0$$ of a sheaf on $$\mathbb {P}^1$$ , landing in the compactly supported completed $$\mathbb {C}_p$$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $$1$$ . For classical weight $$k\geq 2$$ , the Verma has an algebraic quotient $$H^1(\mathbb {P}^1, \mathcal {O}(-k))$$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $$\mathbb {P}^1$$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.
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- Award ID(s):
- 1704005
- PAR ID:
- 10300248
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 9
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2021.16
