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arXiv · 2501.03369

Arithmetic genus inequalities with an application to sums of squares

David Grimm·Gonzalo Manzano-Flores
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Abstract

We show variants of the genus inequality for the irreducible components of the special fiber of an arithmetic curve over a henselian discrete valuation ring of residue characteristic zero that take into account the non-existence of rational, respectively real points on the the components. We then apply this inequality to obtain the bound $2^{ng}$ (respectively $2^{n(g+1)}$) on the totally positive sum-of-two-squares index in the function field of a curve of genus $g$ over the field of $n$-fold iterated real Laurent series with (respectively without) real points. The bound $2^{n(g+1)}$ had been previously known only for hyperelliptic curves.

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David Grimm, Gonzalo Manzano-Flores. 2025-01-06. Arithmetic genus inequalities with an application to sums of squares. https://arxiv.org/abs/2501.03369

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