Barry Green

Realizing deformations of curves using Lubin-Tate formal groups

Letk be an algebraically closed field of characteristicp>0 andR be a suitable valuation ring of characteristic 0, dominating the Witt vectorsW(k). We show how Lubin-Tate formal groups can be used to lift those orderpn automorphisms ofk[[Z]] toR[[Z]], which occur as endomorphisms of a formal group overk of suitable height. We apply this result to prove the existence of smooth liftings of galois covers of smooth curves from characteristicp to characteristic 0, provided thep-part of the inertia groups acting on the completion of the local rings at the points of the cover overk arep-power cyclic and determined by an endomorphism of a suitable formal group overk.

On the small and essential ideals in certain classes of rings

It is well known that for a ring with identity the Brown-McCoy radical is the maximal small ideal. However, in certain subrings of complete matrix rings, which we call structural matrix rings, the maximal small and minimal essential ideals coincide. In this paper we characterize a class of commutative and a class of non-commutative rings for which this coincidence occurs, namely quotients of Prufer domains and structural matrix rings over Brown-McCoy semisimple rings. A similarity between these two classes is obtained.

On the Riemann-Roch theorem for orders in the ring of valuation vectors of a function field

In 1953 Kenkichi Iwasawa, following a suggestion of Artin, gave a characterisation of the ring of valuation vectors (also called repartitions) for function fields in simple topological algebraic terms. Using elementary properties of these rings a short and elegant proof of the Riemann-Roch theorem for smooth complete curves was given. In this paper the methods of linear topology and duality are used to study the Riemann-Roch problem for algebraic curves with singularities. Accordingly we study the linearly compact open modules associated with certain subrings of the ring of valuation vectors of the function field. By applying these methods the Riemann-Roch theorem for algebraic curves with singularities is extended to a larger class of modules than was usual in the literature.

Geometric families of constant reductions and the Skolem property

Let F |K be a function field in one variable and V be a family of independent valuations of the constant field K. Given v ∈ V , a valuation prolongation v to F is called a constant reduction if the residue fields Fv|Kv again form a function field of one variable. Suppose t ∈ F is a non-constant function, and for each v ∈ V let Vt be the set of all prolongations of the Gauß valuation vt on K(t) to F. The union of the sets Vt over all v ∈ V is denoted by V t. The aim of this paper is to study families of constant reductions V of F prolonging the valuations of V and the criterion for them to be principal, that is to be sets of the type V t. The main result we prove is that if either V is finite and each v ∈ V has rational rank one and residue field algebraic over a finite field, or if V is any set of non-archimedean valuations of a global field K satisfying the strong approximation property, then each geometric family of constant reductions V prolonging V is principal. We also relate this result to the Skolem property for the existence of V-integral points on varieties over K, and Rumely’s existence theorem. As an application we give a birational characterization of arithmetic surfaces X/S in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite

ON THE IMMEDIATE EXTENSIONS OF A VALUED FIELD

Abstract In this paper the topological methods introduced by Kaplansky and the theory of linear compactifications are used to prove a result classifying the maximal immediate extensions of a valued field. Results on the existence of a complete discrete rank n valued field of characteristic 0 with prescribed residue class field of characteristic p > 0 are discussed. By applying results of Endler and Ribenboim the existence of a valued field of characteristic 0 and having prescribed residue field of characteristic p > 0 when the value group has finite rank but need not be discrete is demonstrated.

African Institute for the Mathematical Sciences

Meru Alagalingam, Herrenberg Prof. Dr. Martin Arnold, Halle Sebastian Banert, Chemnitz apl.Prof. Dr. Günter Bärwolff, Berlin Sebastian Basten, Heidelberg Stefan Bauer, Regen Johannes Gerd Becker, Zürich Timo Berthold, Berlin Dr.h.c. Gerd Biegel, Braunschweig Stefan Blei, Jena Burkhard Blobel, Quedlinburg Hilmar Böhm, Hamburg Prof. Dr. Malte Braack, Kiel Dr. Markus Brede, Zierenberg Stanislav Bulygin, Kaiserslautern Michael Burger, Beckingen Marc Charpentier, Esch/Alzette Pascal Cremer, Korschenbroich Andreas Decker, Vechta Helge Dietert, Hildesheim Erich Eckner, Tanna Moritz Egert, Darmstadt Nils-Edvin Enkelmann, Ilmenau Daniel Benjamin Fall, Germering Friedrich Feuerstein, Heidelberg Dr. Peter Fiebig, Freiburg Holger Fink, Regensburg Jessica Fintzen, Quickborn Franziska Flegel, Zörbig Dr. Hermann Gebing-Grothus, Bad Vilbel Linda Gründken, Cambridge Magdalena Grüttner, Regensburg Andreas Günther, Hamburg Martin Günther, Bruchsal-Untergrombach Christian Gutschwager, Hannover Jens Hansen, Aachen Markus Hanzig, Wildau Christian Hartfeldt, Magdeburg Shuichi Hayashida, Siegen Torsten Heck, Würselen Sven Herrmann, Darmstadt Peter Hintz, Kaufungen Michael Högele, Berlin Thomas Horneber, Ansbach Dr. Volker Hösel, München Tobias Huxol, Staßfurt Ivan Izmestiev, Berlin Michael Joachim, Münster Adem Kahriman, Albstadt Prof. Dr. Jan Kallsen, Garching Felix Kaschura, Lohsa OT Friedrichsdorf Dr. Dieter Keim, Hagen Anita Kettemann, Stuttgart Peter Kleisinger, Hörsingen Dr. Stefan Krömer, Augsburg Bernhard Christ Kübler, Neubiberg Niklas Kulke, Lehrte Malte Lackmann, Bordesholm Bernd Joachim Laubinger, Burgoberbach Jörg Lehnert, Gelnhausen-Huila Martin Lüders, Schwielowsee/Wildpark-Wes Mareike Massow, Berlin Stefan Mehner, Iserlohn Florian Meier, Maxhütte-Haidhof Prof. Dr. Andreas Meister, Kassel Johannes Muhle-Karbe, München Florentin Münch, Jena Gregor Myrach, Berlin Joachim Oberlinger, Coburg Matthias Ohst, Burg André Oppitz, Floß Ulf Panten, Höhr-Grenzhausen Ulrich Pennig, Göttingen Francoise Pescatore, Esch/Alzette Jörg Philipps, Oldenburg Manuel Plate, Wiesloch Dr. Anca Popa-Fischer, Flensburg Henning Pöttker, Alfhausen Björn Reuper, Herzberg Christina Roeckerath, Aachen Lisa Sauermann, Dresden Johann Sawatzky, Glückstadt Peat Schmolke, Berlin Julian Schnidder, Erlangen Georg Schröter, Dresden Reinhard Steffens, Frankfurt Jan Joachim Tessarz, Bangor Stefan Toman, Petershagen Stephan Trenn, Ilmenau Hella Ußler, Rostock Mathias Vetter, Bochum Alexander Wapenhans, Neuenhagen Johannes Warnecke, Münster Philipp Weiß, Hoyerswerda David Willimzig, Schwerin Diether Wittek, Kirchlinteln Hendrik Wöhrmann, Lohne