David Grimm: Sums of three squares in function fields of conics and Cassels-Catalan curves


Submission: 2012, Apr 2

(This preprint is an improved and ouverhauled version of an earlier preprint (No. 306) on this server.) It was shown by Tikhonov and Yanchevskii that the function field of a smooth conic with a real point over a hereditarily pythagorean field has pythagoras number two, and that the same is true for conics without real point over hereditarily euclidean fields. We show that these are in fact the only situations when the pythagoras number of a function field of genus zero can be two. We partly extend this observation to Cassels-Catalan curves.

2010 Mathematics Subject Classification: 12D15, 14H05, 14H45, 14G05

Keywords and Phrases: Pythagoras number, function fields, rational points

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