Arithmetic genus inequalities with an application to sums of squares
David Grimm, Gonzalo Manzano-Flores
TL;DR
This work develops arithmetic-genus inequalities for arithmetic curves over henselian DVRs of residue characteristic zero, incorporating arithmetic restrictions from rational/real points via a refined dual-graph and Galois-symmetry analysis. It combines Raynaud’s cohomological flatness with a normal-crossings framework to derive genus bounds that feed into sharp bounds for sums of squares in function fields over iterated real Laurent series, yielding the optimal real bound $2^{ng}$ and the nonreal bound $2^{n(g+1)}$ (the latter previously known only for hyperelliptic curves). The methodology blends graph-theoretic symmetry, regular-model degeneration, and local-global principles for quadratic forms to also bound local squares in terms of dual-graph genus data, culminating in a unified approach that extends prior results and clarifies the role of residue-field arithmetic. These results have implications for sums-of-squares decompositions and the Kaplansky radical in function fields, and provide sharp, optimal bounds across real and nonreal cases.
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.