Moduli for rational genus 2 curves with real multiplication for discriminant 5
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910915681583104 |
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| author | Cowan, Alex Martin, Kimball |
| author_facet | Cowan, Alex Martin, Kimball |
| contents | Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their Jacobians, but the converse is not true. We give a simple generic description of which rational moduli points correspond to rational curves, as well as give associated Weierstrass models, in the case of RM by the ring of integers of $\mathbb{Q}(\sqrt{5})$. To prove this, we provide some techniques for reducing quadratic forms over polynomial rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_05752 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Moduli for rational genus 2 curves with real multiplication for discriminant 5 Cowan, Alex Martin, Kimball Number Theory Algebraic Geometry 11F41, 11G10, 11G30, 14H10, 14K10, 11E12 Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their Jacobians, but the converse is not true. We give a simple generic description of which rational moduli points correspond to rational curves, as well as give associated Weierstrass models, in the case of RM by the ring of integers of $\mathbb{Q}(\sqrt{5})$. To prove this, we provide some techniques for reducing quadratic forms over polynomial rings. |
| title | Moduli for rational genus 2 curves with real multiplication for discriminant 5 |
| topic | Number Theory Algebraic Geometry 11F41, 11G10, 11G30, 14H10, 14K10, 11E12 |
| url | https://arxiv.org/abs/2206.05752 |