23 research outputs found
Torsion group schemes as iterative differential Galois groups
We are considering iterative derivations on the function field L of abelian
schemes in positive characteristic p>0, and give conditions when the torsion
group schemes of this abelian scheme occur as ID-automorphism groups, i.e. are
the ID-Galois groups of L over certain ID-subfields. For an explicit example,
we even give a construction of (a family of) such iterative derivations.Comment: 14 pages; v2: rewritten some proofs for better readabilit
Galois theory for iterative connections and nonreduced Galois groups
This article presents a theory of modules with iterative connection. This
theory is a generalisation of the theory of modules with connection in
characteristic zero to modules over rings of arbitrary characteristic. We show
that these modules with iterative connection (and also the modules with
integrable iterative connection) form a Tannakian category, assuming some nice
properties for the underlying ring, and we show how this generalises to modules
over schemes. We also relate these notions to stratifications on modules, as
introduced by A. Grothendieck in order to extend integrable (ordinary)
connections to finite characteristic. Over smooth rings, we obtain an
equivalence of stratifications and integrable iterative connections.
Furthermore, over a regular ring in positive characteristic, we show that the
category of modules with integrable iterative connection is also equivalent to
the category of flat bundles as defined by D. Gieseker. In the second part of
this article, we set up a Picard-Vessiot theory for fields of solutions. For
such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes
into account even nonreduced closed subgroup schemes of the Galois group scheme
on one hand and inseparable intermediate extensions of the Picard-Vessiot
extension on the other hand. Finally, we compare our Galois theory with the
Galois theory for purely inseparable field extensions.Comment: 37 pages; v2->v3: more cross references to other papers are added in
this version, the introduction is more detailed v3->v4: proof of Thm. 11.5iv)
and of the following corollaries changed and hyperref adde
Prolongations of t-motives and algebraic independence of periods
In this article we show that the coordinates of a period lattice generator of
the -th tensor power of the Carlitz module are algebraically independent, if
is prime to the characteristic. The main part of the paper, however, is
devoted to a general construction for -motives which we call prolongation,
and which gives the necessary background for our proof of the algebraic
independence. Another ingredient is a theorem which shows hypertranscendence
for the Anderson-Thakur function , i.e. that and all its
hyperderivatives with respect to are algebraically independent.Comment: 21 pages; v1->v2: extended the basic notation for better readability,
corrected typos; final version to appear in Documenta Mathematic
Reduced group schemes as iterative differential Galois groups
This article is on the inverse Galois problem in Galois theory of linear
iterative differential equations in positive characteristic. We show that it
has an affirmative answer for reduced algebraic group schemes over any
iterative differential field which is finitely generated over its algebraically
closed field of constants. We also introduce the notion of equivalence of
iterative derivations on a given field - a condition which implies that the
inverse Galois problem over equivalent iterative derivations are equivalent.Comment: 13 page