Deformation spaces of one-dimensional formal modules and their cohomology

Let Mm be the formal scheme which represents the functor of deformations of a one-dimensional formal module over equipped with a level-m-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the ?-adic étale cohomology of the generic fibre Mm of Mm realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces Mm.     

Links between associated additive Galois modules and computation of H for local formal group modules

Using the methods described in the papers (Documenta Math. 5 (2000) 657; Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields, to appear), we prove that a cocycle for a formal group in a Galois p-extension of a complete discrete valuation field is a coboundary if and only if the corresponding group algebra elements increase valuations by a number that is sufficiently large. We also calculate the valuation of the splitting element of a coboundary. A special case of the main theorem allows us to determine when a p-extension of a complete discrete valuation fields contains a root of a Kummer equation for a formal group. The theorem of Coates-Greenberg for formal group modules in deeply ramified extensions is generalized to noncommutative formal groups. Some results concerning finite torsion modules for formal groups are obtained.     

Decomposing torsion modules

E. Matlis proved that if R is an integral domain with quotient field Q and K is the R-module Q/R, then all torsion R-modules decompose into a direct sum of local submodules if and only if K decomposes into a direct sum of local submodules. Thus K is a test module to determine whether torsion modules decompose. We generalize this result to commutative rings. If R is a commutative ring and a torsion theory of R is given by a Gabriel topology , then form the ring of quotients R_ and let K be the cokernel of the canonical ring homomorphism from R to R_. In some special cases, every -torsion R-module decomposes into a direct sum of local submodules if and only if K decomposes. However, there is an example where this is not the case. The principal result is: given R,  and K, there is a related filter _K of ideals of R, which is a subset of , such that all _K-pretorsion R-modules decompose into a direct sum of local submodules if and only if K decomposes. The relationship between  and _K is investigated.     

Galois module structure of Tate modules

Dedicated to Professor Peter Roquette Received 1 February 1996; in final form 26 April 1996     

Galois actions on Q-curves and winding quotients

We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free. The first author is supported by BFM2003-06092.     

Partial actions and Galois theory

In this article, among other results, we develop a Galois theory of commutative rings under partial actions of finite groups, extending the well-known results by S.U. Chase, D.K. Harrison and A. Rosenberg.     

Gluing torsion endo-permutation modules

Let k be a field of characteristic p, and let P be a finitep-group, where p is an odd prime. In this paper, we considerthe problem of gluing compatible families of endo-permutationmodules: being given a torsion element M_Q in the Dade groupD(N_P(Q)/Q), for each non-trivial subgroup Q of P, subject toobvious compatibility conditions, we show that it is alwayspossible to find an element M in the Dade group of P such that for all Q, but that Mneed not be a torsion element of D(P). The obstruction to thisis controlled by an element in the zeroth cohomology group over₂ of the poset of elementary abelian subgroups of P of rankat least 2. We also give an example of a similar situation,when M_Q is only given for centric subgroups Q of P. Moreover,general results about biset functors and the Dade functor aregiven in two appendices.     

Galois modules in a multidimensional local field

The structure of the group of principal units is studied as a Galois module in a weakly ramified extension of a multidimensional local field. A criterion is given for the existence of a normal basis for additive Galois modules in such extensions. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 198, pp. 5–14, 1991.     

Primary elements in formal modules

The stability of an elastic plate in a supersonic gas flow is considered in the presence of a boundary layer formed on the surface of the plate. The problem is solved in two statements. In the first statement, the plate is of large but finite length, and a coupled-mode type of flutter is examined (the effect of the boundary layer on another, single-mode, type of flutter has been studied earlier). In the second statement, the plate is assumed to be infinite, and the character of its instability (absolute or convective) is analyzed. In both cases, the instability is determined by a branch point of the roots of the dispersion equation, and the mathematical analysis is the same. It is proved that instability in a uniform gas flow is weakened by a boundary layer but cannot be suppressed completely, while in the case of a stable plate in a uniform flow the boundary layer leads to the destabilization of the plate.     

Relative Lubin-Tate formal groups and Galois module structure

Some Galois modules attached to divisions fields of relative Lubin-Tate formal groups are studied. Their associated order and module generators are determined.     

Decomposing finitely generated torsion modules

Let R be a commutative ring and let F be a filter of ideals of R. We give three characterizations of the rings with the property that all finitely generated F pretorsion R modules decompose into a direct sum of cyclic R submodules. One of these characterizations involves decomposing the filter F into a product of subfilters. This filter decomposition is shown to be unique, and is used to characterize several other ring properties relative to the filter of ideals.