Timeline for Canonical lifts from $\mathbb F_q$ and CM-theory
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2014 at 19:53 | vote | accept | Calodeon | ||
| Apr 13, 2014 at 16:19 | answer | added | answer_bot | timeline score: 3 | |
| Apr 11, 2014 at 0:17 | comment | added | Calodeon | Thanks ! I don't think I understand how exactly there is an equivalence of categories between ordinary abelian varieties over a finite field and over $\mathbb C$ in Deligne's paper though... I would expect a result like, fixing an embedding $W(k) \rightarrow \mathbb C$, every variety over $k$ of fixed endomorphism ring $\mathcal O$ lifts to a curve $\mathbb C^n/\Lambda(\mathfrak a)$ (up to isomorphism), and reciprocally every such complex curve reduces well (reduction by a prime over $p$ in a number field $L$ ?) back to $k$. (and isogenies would also lift and reduce well...). | |
| Apr 10, 2014 at 21:33 | comment | added | stankewicz | I have two references in mind: 1) Messing's book shows how if $A_{/k}$ is an ordinary abelian variety then the Serre-Tate Canonical lift $\mathcal{A}_{W(k)}$ admits an isomorphism $End_k(A) \cong End_{W(k)}(\mathcal{A})$. 2) Deligne's Inventiones paper "Varieties Abeliennes Ordinaires..." shows how the choice of embedding $W(k) \hookrightarrow \mathbf{C}$ doesn't change anything. He therefore shows an equivalence of categories between ordinary abelian varieties over a finite field vs $\mathbf{C}$ whose details escape me at the moment but the paper is 3 pages so you should read it anyway :) | |
| Apr 10, 2014 at 21:19 | history | edited | Calodeon | CC BY-SA 3.0 |
added 26 characters in body
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| Apr 10, 2014 at 6:28 | history | asked | Calodeon | CC BY-SA 3.0 |