Timeline for showing that abelian varieties are de Rham *without* showing that they are crystalline
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 16, 2014 at 7:54 | answer | added | gyr | timeline score: 1 | |
| Jan 23, 2013 at 9:49 | comment | added | anon | In the case of abelian varieties with good reduction (or even p-divisible groups), Faltings give a direct proof (avoiding the machinery of almost etale extensions, close in spirit to Fontaine's proof) in his "Integral crystalline cohomology over very ramified rings" paper. | |
| Jan 23, 2013 at 8:51 | comment | added | ACL | Correction : "For abelian varieties with good reduction, an extension..." | |
| Jan 23, 2013 at 8:51 | comment | added | ACL | An extension to more general base rings than an $p$-adic field is shown in a paper by Jean-Pierre Wintenberger, ams.org/mathscinet-getitem?mr=1293978, to be found in the volume 223 of Astérisque, ``Périodes $p$-adiques'' (edited by Fontaine). I would bet than one can obtain the more general result along the same lines (at least if the ab. var. has semi-stable reduction over $K$) by adding as an input results of Raynaud in the same volume, or uses the universal vector extension of (the connected component of) a Néron model (Mazur-Messing). | |
| Jan 23, 2013 at 8:28 | comment | added | user28172 | What you attribute to Faltings is stated incorrectly: you need to either weaken the "crystalline" condition or assume stronger "good reduction" hypotheses on $X$. (Likewise, Fontaine's result was not for "general" abelian varieties, but rather those with good reduction, studied via their $p$-divisible groups over the valuation ring.) | |
| Jan 23, 2013 at 4:00 | history | asked | Tony | CC BY-SA 3.0 |