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 Title: Algebraic Curves Over Supersimple Fields Author(s): Martin-Pizarro, Amador Doctoral Committee Chair(s): Pillay, Anand Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: 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. Issue Date: 2003 Type: Text Language: English Description: 76 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003. URI: http://hdl.handle.net/2142/86828 Other Identifier(s): (MiAaPQ)AAI3111578 Date Available in IDEALS: 2015-09-28 Date Deposited: 2003
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