Publications and Submitted Preprints

• Non-archimedean hyperbolicity and applications (joint with A. Javanpeykar)
  (Submitted) We introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties. We use it to show that if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application, we show that the moduli space of abelian varieties is analytically Brody hyperbolic in equal characteristic zero. Finally, we use Scholze's uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the ``Theorem of the Fixed Part'' in mixed characteristic. arXiv.

• Rigidity for rigid analytic motives (joint with F. Bambozzi)
  (J. Inst. Math. Jussieu (2019), 1--29 ) We extend the Rigidity Theorem to motives of rigid analytic varieties over a non-Archimedean valued field. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of the étale realization functors and the improvement of the comparison between motives with and without transfers as well as the rigid analytic motivic tilting equivalence. arXiv, article.

• The Berkovich realization for rigid analytic motives
  (J. Algebra 527 (2019), 30--54) We prove that the functor associating to a rigid analytic variety the singular complex of the underlying Berkovich topological space is motivic, and defines the maximal Artin quotient of a motive. We use this to generalize Berkovich's results on the weight-zero part of the étale cohomology of a variety defined over a non-archimedean valued field. arXiv, article.

• Rigid cohomology via the tilting equivalence
  (J. Pure Appl. Algebra 223 (2019), no. 2, 818--843) We define a de Rham cohomology theory for analytic varieties over a valued field of equal characteristic p with coefficients in a chosen untilt of its perfection by means of the motivic version of Scholze's tilting equivalence. We show that this definition generalizes the usual rigid cohomology in case the variety has good reduction. We also prove a conjecture of Ayoub yielding an equivalence between rigid analytic motives with good reduction and unipotent algebraic motives over the residue field. arXiv, updated version, article.

• A Motivic Version of the Theorem of Fontaine and Wintenbrger
  (Compos. Math. 155(1) (2019), 38-88) We prove the equivalence between the categories of motives of rigid analytic varieties over a perfectoid field of mixed characteristic and over the associated (tilted) perfectoid field of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of the two fields are isomorphic. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces. arXiv, pdf, article.

• The Monsky-Washnitzer and the overconvergent realizations
  (Int. Math. Res. Not. 11 (2018), 3443-3489) We outline the constuction of the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classic étale site provides a new direct definition of rigid cohomology and shows that its finite dimensionality follows formally from the one of Betti cohomology for smooth projective complex varieties. arXiv, updated version, article.

• Effective Motives with and without Transfers in characteristic p
  (Adv. Math. 306 (2017), 852-879) We prove the equivalence between the category of effective étale motives with transfers of rigid analytic varieties over a perfect complete non-archimedean field and the category of Frob-étale motives, obtained by localizing the category of motives without transfers over purely inseparable maps. In particular, we obtain an equivalence between motives with and without transfers in the characteristic 0 case and an equivalence between étale with transfers and Frob-étale motives of smooth algebraic varieties over a perfect field. We also show a relative and a stable version of the main statement. arXiv, article.

• Deitmar's Versus Töen-Vaquié's Schemes Over F_1
  (Math. Z., 271, Issue 3 (2012), 911-926) We prove the equivalence of Deitmar's and Töen-Vaquié's definitions of schemes over the "field with one element", checking compatibility with base change to schemes over Z. Using the same methods, we give a new proof of the equivalence of the "functorial" and the "geometrical" definitions of schemes. arXiv, article.

Theses

• A Motivic Version of the Theorem of Fontaine and Wintenbrger
  (PhD Thesis) We prove the equivalence between the categories of motives of rigid analytic varieties over a perfectoid field of mixed characteristic and over the associated (tilted) perfectoid field of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of the two fields are isomorphic. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces and a version of the equivalence between motives with and without transfers in positive characteristic, which is also proven. pdf.

• The Geometry Over the Field with One Element
  (Master's Thesis) We introduce the theory of schemes over F_1 and prove the equivalence of Deitmar's and Töen-Vaquié's definitions. We also compute the Picard group of P^n and study the base change functor on Pic. Appendices with motivations, basics of stack theory and the theory of differentials for monoids, following Kato. pdf, summary(ita).

A Selection of Research Talks

• Realization functors for rigid analytic motives
  (Algebraic K-theory and Arithmetic, Bedlewo, 2017)

• The tilting equivalence and motivic Galois groups
  (Paris-London Number Theory Seminar, 2016)

• L'equivalenza di tilting rigido-analitica motivica
  (20th Congress of the Italian Mathematical Society, 2015)

• Le basculement analytique rigide motivique
  (SAGA Université Paris 11, 2015)

• Le basculement analytique rigide motivique
  (Séminaire Autour des Cycles Algébriques, Paris, 2014)

• The motivic rigid analytic tilting equivalence
  (AriVaF conference, Bordeaux, 2014)

• A geometrical and a functorial approach to schemes over F_1
  (CIRM Luminy, 2011)

A Selection of Seminar Talks

• Omega-hat and p-adic uniformization
  (ETH ProDoc Workshop at Alpbach, 2015) website.

• Groupes analytiques rigides p-divisibles
  (Groupe de travail, IRMAR Rennes, 2015) website.

• Commutative formal groups, Tate's results
  (ETH ProDoc Workshop at Alpbach, 2014) website.

• Rigid modular curves
  (ETH ProDoc Workshop at Alpbach, 2013) website.

• The Pro-Unipotent Completion
  (ETH ProDoc Workshop at Alpbach, 2012) website, pdf.

• The Jacquet-Langlands correspondence
  (Universität Zürich, 2013)

• Non-cuspidal representations of GL(2,Q_p)
  (Universität Zürich, 2012)

• Adic spaces and the acyclicity theorem
  (Università degli Studi di Milano, 2011)

• Universal Homotopy Theories
  (Summer School in Brixen, 2011)

• A Functorial Characterization of Open Immersions
  (University of Leiden, 2009) pdf.

• Monoidal Model Categories and Modules over Them
  (Università degli Studi di Milano, 2009)