## Number Theory Seminar

Seminar information archive ～09/10｜Next seminar｜Future seminars 09/11～

Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Naoki Imai, Shane Kelly |

### 2008/10/29

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Overholonomicity of overconvergence $F$-isocrystals on smooth varieties

**Daniel Caro**(Université de Caen)Overholonomicity of overconvergence $F$-isocrystals on smooth varieties

[ Abstract ]

Let $¥mathcal{V}$ be a complete discrete valuation ring

of characteristic $0$, with perfect residue field $k$ of

characteristic $p>0$. In order to construct $p$-adic coefficients

over $k$-varieties, Berthelot introduced the theory of

overconvergent $F$-isocrystals, i.e overconvergent isocrystals with

Frobenius structure. Moreover, to get a $p$-adic cohomology over

$k$-varieties stable under cohomological operations, Berthelot built

the theory of arithmetic $F$-$¥mathcal{D}$-modules. In this talk,

after recalling some elements of these theories, we introduce the

notion of overholonomicity with is a property as stable as the

holonomicity in the classical theory of $¥mathcal{D}$-modules. The

goal of the talk is to prove the overholonomicity of arithmetic

$¥mathcal{D}$-modules associated to overconvergent $F$-isocrystals

over smooth $k$-varieties. In the proof we need Christol's transfert

theorem, a comparison theorem between relative log rigid cohomology

and relative rigid cohomology and last but not least Kedlaya's

semistable reduction theorem. This is a joint work with Nobuo

Tsuzuki.

Let $¥mathcal{V}$ be a complete discrete valuation ring

of characteristic $0$, with perfect residue field $k$ of

characteristic $p>0$. In order to construct $p$-adic coefficients

over $k$-varieties, Berthelot introduced the theory of

overconvergent $F$-isocrystals, i.e overconvergent isocrystals with

Frobenius structure. Moreover, to get a $p$-adic cohomology over

$k$-varieties stable under cohomological operations, Berthelot built

the theory of arithmetic $F$-$¥mathcal{D}$-modules. In this talk,

after recalling some elements of these theories, we introduce the

notion of overholonomicity with is a property as stable as the

holonomicity in the classical theory of $¥mathcal{D}$-modules. The

goal of the talk is to prove the overholonomicity of arithmetic

$¥mathcal{D}$-modules associated to overconvergent $F$-isocrystals

over smooth $k$-varieties. In the proof we need Christol's transfert

theorem, a comparison theorem between relative log rigid cohomology

and relative rigid cohomology and last but not least Kedlaya's

semistable reduction theorem. This is a joint work with Nobuo

Tsuzuki.