The project is about the study of two groups which are naturally assosciated to a field and which contain different information on the arithmetic of the field, related to quadratic form theory. One is the quotient of the group of nonzero sums of squares modulo the subgroup of sums of two squares, the other one is the so-called Kaplansky radical.
These two groups are studied in a context of arithmetic geometry, namely for algebraic function fields over a complete discretely valued field. Such a field can be viewed as the function field of an arithmetic surface over a discrete valuation ring. Classical reduction theory of algebraic curves relates our problem to the study of the special fiber of this arithmetic surface, which is an algebraic curve over the residue field.
The reduction curve is connected, and the analysis of the genera of its irreducible components and of the intersections of these components relates our problems (on the two specific groups) to a combinatorial problem on the associated intersection graph.
The aim of the PhD project of Gonzalo Manzano Flores in this context is in particular to obtain interesting examples of curves which illustrate how big the two groups can be under given conditions on the genus of the function field.
We are searching for such examples over very specific complete discretely valued fields, namely Laurent series fields in one variable over the real and over the complex numbers, R((t)) and C((t)).
The PhD candidate has obtained results on the first group (sums of squares modulo sums of two squares) in the case of a function field of a hyperelliptic curve over the field R((t)).
These examples show that the upper bound for the size of this group obtained in Becher, Van Geel, Sums of squares in function fields of hyperelliptic curves. Math. Z. 261 (2009): 829–844, is best possible in the case of the function field of a hyperelliptic curve of any even degree with a nonreal function field.
On the other hand, Gonzalo Manzano Flores also found evidence that this bound can be slightly improved in the case of a real function field, and that the improved bound then is optimal. The PhD candidate is currently writing up these results, which will cover a substantial part of his thesis.
Complementary to this topic, we started in 2019 to work on the Kaplansky radical for function fields of arithmetic surfaces, and obtained first results and additionally evidence for some intriguing relations to the other problem.
The Kaplansky radical was introduced in the 1970's by C. Cordes as a way to generalise a couple of results in quadratic form theory by substituting this group for the subgroup of squares in the field. However, it took a while until examples of fields with a nontrivial Kaplansky radical were discovered, in the 1980's by M. Kula.
In the last two decades, a series of deep results on local-global principles have come up in quadratic form theory, which hold over function fields of arithmetic surfaces, and it has turned out that there is often in these situations a failure of the local-global principle in dimension 2. It was described in Becher, Leep, The Kaplansky radical of a quadratic field extension, Journal of Pure and Applied Algebra 218 (2014), 1577-1582 that this failure in dimension 2 is directly expressed by the Kaplansky radical of the corresponding function field.
The aim of the collaboration with Gonzalo Manzano Flores and his supervisor at USACH, prof. David Grimm, during the period of funding will be to complement this observation by a variety of generic examples, in particular for hyperelliptic curves over C((t)).
These results together with the topic on sums of squares shall lead in the academic year 2020/21 to a completion of the PhD of Gonzalo Manzano Flores, according the dubble degree convention between USACH and UA, with myself and prof. David Grimm as promotors.