Etale and crystalline companions, I
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2018
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914204469952512 |
|---|---|
| author | Kedlaya, Kiran S. |
| author_facet | Kedlaya, Kiran S. |
| contents | 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 étale 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 étale 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 étale coefficient object has étale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has étale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of étale coefficient objects; this subject will be pursued in a subsequent paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_00204 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Etale and crystalline companions, I Kedlaya, Kiran S. Number Theory Algebraic Geometry 14F30, 14F20 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 étale 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 étale 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 étale coefficient object has étale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has étale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of étale coefficient objects; this subject will be pursued in a subsequent paper. |
| title | Etale and crystalline companions, I |
| topic | Number Theory Algebraic Geometry 14F30, 14F20 |
| url | https://arxiv.org/abs/1811.00204 |