Galois actions on torsion points of universal one-dimensional formal modules
نویسنده
چکیده
Let F be a local non-Archimedean field with ring of integers o and uniformizer ̟. Let X be a one-dimensional formal o-module of F -height n over the algebraic closure F of the residue field of o. By the work of Drinfeld, the universal deformation X of X is a formal group over a power series ring R0 in n − 1 variables over the completion of the maximal unramified extension ônr of o. For h ∈ {0, . . . , n − 1} let Uh ⊂ Spec(R0) be the locus where the connected part of the associated ̟-divisible module X[̟∞] has height h. Using the theory of Drinfeld level structures we show that the representation of π1(Uh) on the Tate module of the étale quotient is surjective.
منابع مشابه
ALGEBRAS WITH CYCLE-FINITE STRONGLY SIMPLY CONNECTED GALOIS COVERINGS
Let $A$ be a nite dimensional $k-$algebra and $R$ be a locally bounded category such that $R rightarrow R/G = A$ is a Galois covering dened by the action of a torsion-free group of automorphisms of $R$. Following [30], we provide criteria on the convex subcategories of a strongly simply connected category R in order to be a cycle- nite category and describe the module category of $A$. We p...
متن کاملTorsion bounds for elliptic curves and Drinfeld modules
We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L : K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Rütsche, respectively. Our approach...
متن کاملReport on Research in Teams Project: Universal Higher Extensions
In one stream of development, one partner in the project had just achieved an interpretation of group cohomology or Lie-algebra cohomology in terms of higher-dimensional central extensions, as developed by Rodelo–Van der Linden [14, 15]. It extends the classical interpretation of the second group cohomology in terms of equivalence classes of short exact sequences with central kernel. This devel...
متن کاملON SYMPLECTIC ISOMORPHIMS OF THE p-TORSION OF ELLIPTIC CURVES
Let ` and p be distinct prime numbers with p C 3. Let E~Q` and E~Q` be elliptic curves de ned over Q`, having potentially good reduction with a defect of semistability of order 3. We suppose that the Galois modules E p and E p of their p-torsion points are isomorphic, and that their eld of p-torsion points is non-abelian over Q`. We establish a criterion, in terms of the standard invariants as...
متن کاملThe Galois theory of orbits in arithmetic dynamics
Arboreal Galois groups sit naturally as subgroups of tree (or graph) automorphism groups, while dynatomic Galois groups are naturally subgroups of certain wreath products. A fundamental problem is to determine general conditions under which these dynamically generated Galois groups have finite index in the natural geometric groups that contain them. This is a dynamical analog of Serre’s theorem...
متن کامل