Arithmetic genus inequalities with an application to sums of squares

TL;DR

This paper develops new inequalities related to the genus of arithmetic curves, incorporating the absence of rational or real points, and applies these to bound sums of squares indices in function fields over iterated real Laurent series.

Contribution

It introduces variants of the genus inequality that account for point non-existence and applies them to derive bounds on sums of squares indices in specific function fields.

Findings

Derived bounds on sums of squares indices: $2^{ng}$ and $2^{n(g+1)}$.

Extended known bounds from hyperelliptic to more general curves.

Connected genus inequalities with arithmetic properties of curve components.

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.