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arXiv · 2211.12078

Plectic structures in p-adic de Rham cohomology

David Loeffler·Sarah Livia Zerbes
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Abstract

Given a Hilbert modular form for a totally real field $F$, and a prime $p$ split completely in $F$, the $f$-eigenspace in $p$-adic de Rham cohomology of the Hilbert modular variety has a family of partial filtrations and partial Frobenius maps, indexed by the primes of $F$ above $p$. The general plectic conjectures of Nekovar and Scholl suggest a "plectic comparison isomorphism" comparing these structures to etale cohomology. We prove this conjecture in the case $[F : \mathbf{Q}] = 2$ under some mild assumptions; and for general $F$ we prove a weaker statement which is strong evidence for the conjecture, showing that plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii. (In memory of Jan Nekovar)

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David Loeffler, Sarah Livia Zerbes. 2022-11-22. Plectic structures in p-adic de Rham cohomology. https://doi.org/10.1016/j.jnt.2023.10.016

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