Arithmetic genus inequalities with an application to sums of squares
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866929661520379904 |
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| author | Grimm, David Manzano-Flores, Gonzalo |
| author_facet | Grimm, David Manzano-Flores, Gonzalo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03369 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic genus inequalities with an application to sums of squares Grimm, David Manzano-Flores, Gonzalo Number Theory Algebraic Geometry 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. |
| title | Arithmetic genus inequalities with an application to sums of squares |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2501.03369 |