Etale and crystalline companions, I
Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell \neq p$, and overconvergent $F$-isocrystals in rigid cohomology wh...
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| Format: | Article |
| Language: | English |
| Published: |
Association Epiga
2022-12-01
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| Series: | Épijournal de Géométrie Algébrique |
| Subjects: | |
| Online Access: | https://epiga.episciences.org/6820/pdf |
| _version_ | 1797248738712879104 |
|---|---|
| author | Kiran S. Kedlaya |
| author_facet | Kiran S. Kedlaya |
| author_sort | Kiran S. Kedlaya |
| collection | DOAJ |
| description | Let $X$ be a smooth scheme over a finite field of characteristic $p$.
Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic
Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell
\neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$.
Using the Langlands correspondence for global function fields in both the
\'etale and crystalline settings (work of Lafforgue and Abe, respectively), one
sees that on a curve, any coefficient object in one category has "companions"
in the other categories with matching characteristic polynomials of Frobenius
at closed points. A similar statement is expected for general $X$; building on
work of Deligne, Drinfeld showed that any \'etale coefficient object has
\'etale companions. We adapt Drinfeld's method to show that any crystalline
coefficient object has \'etale companions; this has been shown independently by
Abe--Esnault. We also prove some auxiliary results relevant for the
construction of crystalline companions of \'etale coefficient objects; this
subject will be pursued in a subsequent paper. |
| first_indexed | 2024-04-24T20:19:22Z |
| format | Article |
| id | doaj.art-43c5bdac902a44828f5f98b232982377 |
| institution | Directory Open Access Journal |
| issn | 2491-6765 |
| language | English |
| last_indexed | 2024-04-24T20:19:22Z |
| publishDate | 2022-12-01 |
| publisher | Association Epiga |
| record_format | Article |
| series | Épijournal de Géométrie Algébrique |
| spelling | doaj.art-43c5bdac902a44828f5f98b2329823772024-03-22T09:11:16ZengAssociation EpigaÉpijournal de Géométrie Algébrique2491-67652022-12-01Volume 610.46298/epiga.2022.68206820Etale and crystalline companions, IKiran S. Kedlayahttps://orcid.org/0000-0001-8700-8758Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell \neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$. Using the Langlands correspondence for global function fields in both the \'etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general $X$; building on work of Deligne, Drinfeld showed that any \'etale coefficient object has \'etale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has \'etale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of \'etale coefficient objects; this subject will be pursued in a subsequent paper.https://epiga.episciences.org/6820/pdfmathematics - number theorymathematics - algebraic geometry14f30, 14f20 |
| spellingShingle | Kiran S. Kedlaya Etale and crystalline companions, I Épijournal de Géométrie Algébrique mathematics - number theory mathematics - algebraic geometry 14f30, 14f20 |
| title | Etale and crystalline companions, I |
| title_full | Etale and crystalline companions, I |
| title_fullStr | Etale and crystalline companions, I |
| title_full_unstemmed | Etale and crystalline companions, I |
| title_short | Etale and crystalline companions, I |
| title_sort | etale and crystalline companions i |
| topic | mathematics - number theory mathematics - algebraic geometry 14f30, 14f20 |
| url | https://epiga.episciences.org/6820/pdf |
| work_keys_str_mv | AT kiranskedlaya etaleandcrystallinecompanionsi |