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Abstract We study the cone of moving divisors on the moduli space $${\mathcal{A}}_{g}$$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on $${\mathcal{A}}_{g}$$ for $$g\leq 4$$, and gives an explicit upper bound for the moving slope of $${\mathcal{A}}_{5}$$ and a conjectural upper bound for the moving slope of $${\mathcal{A}}_{6}$$.
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Slope classicality in higher Coleman theory via highest weight vectors in completed cohomology
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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- Award ID(s):
- 2201112
- PAR ID:
- 10410545
- Date Published:
- Journal Name:
- Proceedings of the National Academy of Sciences
- Volume:
- 119
- Issue:
- 45
- ISSN:
- 0027-8424
- 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/10.1073/pnas.2208249119
