A crystalline incarnation of Berthelot's conjecture and K\"unneth formula for isocrystals

Berthelot's conjecture predicts that under a proper and smooth morphism of schemes in characteristic $p$, the higher direct images of an overconvergent $F$-isocrystal are overconvergent $F$-isocrystals. In this paper we prove that this is true for crystals up to isogeny. As an application we prove a K\"unneth formula for the crystalline fundamental group

On p-adic differential equations on semistable varieties

This is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this record.In this paper we prove a comparison theorem between the category of certain modules with integrable connection on the complement of a normal crossing divisor of the generic fiber of a proper semistable variety over a DVR and the category of certain log overconvergent isocystrals on the special fiber of the same open

Kernel of the monodromy operator for semistable curves

This is the final version of the article. Available from the publisher via the URL in this recordThis is an announcement of results, obtained in collaboration with B. Chiarellotto, R. Coleman and A. Iovita, whose proofs will be published elsewhere. We study the relation between the kernel of the monodromy operator acting on the first log crystalline cohomology group of the special fiber of a semistable curve and the first rigid cohomology group with non trivial coefficients

Basi normali intere: risultati principali nel caso generale e sviluppi recenti nel caso abeliano.

Una base normale intera per un'estensione di campi di numeri L/K di Galois è una base dell'anello degli interi di L come modulo sull'anello degli interi di K costituita da elementi coniugati. Trovare una condizione necessaria e sufficiente per l'esistenza di una base normale intera di una data estenisione è un problema aperto. Se L/K è un'estensione di campi locali una condizione necessaria e sufficiente per l'esistenza di una base normale intera è che l'estensione sia tame. Nel caso di campi globali non è noto alcun risultato generale; si sa però che nel caso di estensioni abeliane dei razionali la ramificazione moderata basta a garantire l'esistenza di una tale base. Nell'ultima parte della tesi vengono studiate le estensioni L/K tali che K sia un campo ciclotomico e L un'estensione abeliana dei razionali: viene data una condizione necessaria e sufficiente perché una esista una base normale per queste particolari estensioni e viene definito un indice che misura quanto manca ad una tale estensione all'avere una base normale intera

On p-adic differential equations on semistable varieties II

This is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this record.This paper is a complement to the paper "On p-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequence, we can find suitable Tannakian subcategories of log overconvergent isocrystals and of modules with regular integrable connection on which the algebraization functor is an equivalence of Tannakian categories.The main part of this work was done when the first author was at the Graduate School of Mathematical Sciences of the University of Tokyo supported by a postdoctoral fellowship and kaken-hi (grant-in-aid) of the Japanese Society for the Promotion of Science (JSPS). She is now supported by a postdoctoral fellowship of Labex IRMIA.When the main part of this work was done, the second author was supported by JSPS Grant-in-Aid for Young Scientists (B) 21740003 and Grant-in-Aid for Scientific Research (B) 22340001. Currently he is supported by JSPS Grant-in-Aid for Scientific Research (C) 25400008 and Grant-in-Aid for Scientific Research (B) 23340001

On pp-adic differential equations on semistable varieties II

This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequence, we can find suitable Tannakian subcategories of log overconvergent isocrystals and of modules with regular integrable connection on which the algebraization functor is an equivalence of Tannakian categories

On the homotopy exact sequence for log algebraic fundamental groups

This is the final published version.An earlier version of this paper appears in arXiv and is available in ORE at http://hdl.handle.net/10871/26921The final version is available from Documenta Mathematica via the DOI in this record.We construct a log algebraic version of the homotopy sequence for a quasi-projective normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic