Algebraic Curves Over Supersimple Fields

In this work we are concerned with algebraic curves over supersimple fields. First it is proved that an elliptic curve defined over a supersimple field K whose j-invariant is s-generic over empty has a rational point over K which is s-generic over the set of parameters used to define the curve. Note that a point in a variety over K is s-generic over a given set of parameters A if the SU-rank of the point over A is equal to SU(K) times the dimension of the variety. The same statement as above holds for hyperelliptic curves defined over a supersimple field whose modulus is s-generic over empty . The proof requires a complete description of the smooth models of these curves in all characteristics as well as of the transformations between these curves. Finally, it is shown that for finite Galois extensions of K, the first cohomology group of the Galois group with coefficients in an elliptic curve defined over K is finite. Some similarities and division lines between supersimple fields and other fields with similar cohomological behaviour (for example, C1-fields) are studied. Moreover, a description of quadratic forms over supersimple fields is given.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Martin-Pizarro, Amador

Contributors dc:contributor

• Pillay, Anand

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3111578

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86828