47 research outputs found
On quantum state of numbers
We introduce the notions of quantum characteristic and quantum flatness for
arbitrary rings. More generally, we develop the theory of quantum integers in a
ring and show that the hypothesis of quantum flatness together with positive
quantum characteristic generalizes the usual notion of prime positive
characteristic. We also explain how one can define quantum rational numbers in
a ring and introduce the notion of twisted powers. These results play an
important role in many different areas of mathematics and will also be quite
useful in a subsequent work of the authors.Comment: 2013 - 8
On Quantum Integers and Rationals
ISBN: 978-0-8218-9858-1International audienceWe introduce the notions of quantum characteristic and quantum flatness for arbitrary rings. More generally, we develop the theory of quantum integers in a ring and show that the hypothesis of quantum flatness together with positive quantum characteristic generalizes the usual notion of prime positive characteristic. We also explain how one can define quantum rational numbers in a ring and introduce the notion of twisted powers. These results play an important role in many different areas of mathematics and will also be quite useful in a subsequent work of the authors
Entrevista con Terence Tao, Medalla Fields en el ICM Madrid 2006
8 páginas.-- Dentro de la sección: Las medallas Fields.Peer reviewe
A Simpson correspondence in positive characteristic
International audienceWe define the -curvature map on the sheaf of differential operators of level on a scheme of positive characteristic as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of -curvature on differential modules of level . We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level . Then, using a lifting modulo of Frobenius, we are able to define a Frobenius map on differential operators of level as dual to some divided Frobenius on infinitesimal neighborhhods. We use this map to build a true Azumaya splitting of the completed sheaf of differential operators of level (up to an automorphism of the center). From this, we derive the fact that Frobenius pull back gives, when restricted to quasi-nilpotent objects, an equivalence between Higgs-modules and differential modules of level . We end by explaining the relation with related work of Ogus-Vologodski and van der Put in level zero as well as Berthelot's Frobenius descent
Absolute calculus and prismatic crystals on cyclotomic rings
Let be a prime, the ring of Witt vectors of a perfect field of
characteristic and a primitive th root of unity. We introduce a
new notion of calculus over that we call absolute calculus. It may be seen
as a singular version of the -calculus used in previous work, in the sense
that the role of the coordinate is now played by itself. We show that what
we call a weakly nilpotent absolute connection on a finite free module is
equivalent to a prismatic vector bundle on . As a corollary of a
theorem of Bhatt and Scholze, we finally obtain that an absolute connection
with a frobenius structure on a finite free module is equivalent to a lattice
in a crystalline representation. We also consider the case of de Rham prismatic
crystals as well as Hodge-Tate prismatic crystals