Plectic structures in p-adic de Rham cohomology
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912261231083520 |
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| author | Loeffler, David Zerbes, Sarah Livia |
| author_facet | Loeffler, David Zerbes, Sarah Livia |
| contents | 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) |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_12078 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Plectic structures in p-adic de Rham cohomology Loeffler, David Zerbes, Sarah Livia Number Theory 11F46, 14G35 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) |
| title | Plectic structures in p-adic de Rham cohomology |
| topic | Number Theory 11F46, 14G35 |
| url | https://arxiv.org/abs/2211.12078 |