Sums of squares in function fields over henselian discretely valued fields

2022/03/27 by Manzano-Flores, Gonzalo
#11E04 #11E10 #11E25 #12J10 #14H05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2203.14357

Abstract

Let n∈ℕ and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F/K be an algebraic function field in one variable. We show that the Pythagoras number of F is 2 or 3 and we determine the order of the group of nonzero sums of squares modulo sums of two squares in F in terms of the number of equivalence classes of discrete valuations on F of rank at most n. In the case of function fields of hyperelliptic curves of genus g, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2^n(g+1). We show in Example 4.6 that this bound is optimal.

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