Covers of algebraic varieties and elimination of quantifiers in local fields
by
Konstantin Ponomaryov
Novosibirsk State Technical University

In the paper [1] the author demonstrated his approach to the model theory of local fields. If we have to investigate the structure of definable sets of a local field we have to get a description of images of projections of algebraic sets into projective space. In particular we have to describe images of covers of regular projective varieties. Let us consider a local field of zero characteristic. We can use the resolution of singularities and reduce any finite cover to a cover branched along normal crossing divisor only.

Abhyankar's theory of algebraic fundamental groups forms the basis for study of covers of algebraic varieties. This theory is a powerful tool in solving Zariski problem about covers of projective space branched along normal crossing divisor only [2]. From this point of view the second part of the paper is an easy consequence of affirmative answer to this problem by Abhyankar-Fulton theorem [3].

Now let us consider local fields of nonzero characteristic. Recent progress of M.Spivakovsky in resolution of singularities gives us a chance to repeat arguments above and reduce the investigation of existentially definable sets of any local fields to investigation of covers of a projective space Pⁿ by algebraic variety. But we have a problem in the second stage of our arguments. Indeed, Fulton theorem states solvability of tamely branched cover as above if field has a nonzero characteristic. So we have a problem with wild ramified covers.

As far as any algebraic cover of a regular variety by a normal variety is branched only along some divisor we can use theory of discrete valued fields for elimination of ramification. The recent paper [4] on solvable elimination of ramification in extensions of discrete valued fields gives an opportunity to investigate wild ramification in extensions of fields of any characteristic. Now the author gives a description of existentially definable sets in locally compact fields of any characteristic.

[2]  Zariski O. Algebraic surfaces. Second edition. Springer-Verlag. Berlin, 1971.

[3]  Fulton W. On the fundamental group of the complement of a node curve. Ann. of Math., 111 N2 (1980), 407-409.

[4]  Ponomarev K.N. Solvable elimination of ramification .... Algebra i Logika, 37 N1 (1998), 63-87.

Date received: March 27, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacv-18.