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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2602.14319 |
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| _version_ | 1866915799749361664 |
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| author | Kir, Harun |
| author_facet | Kir, Harun |
| contents | In 1994, Kani introduced an algebraic version of the Humbert invariant, known as the refined Humbert invariant. This invariant q_C is a positive definite quadratic form attached to a smooth curve C of genus 2. It serves as a vital tool, as many geometric properties of C are reflected in the arithmetic properties of q_C. When the Jacobian J_C of a genus 2 curve C is isogenous to a product of an elliptic curve with complex multiplication, the forms q_C have been completely classified recently. In this paper, building upon this classification, we present a constructive algorithm that produces J_C and a divisorial representative of a curve C of genus 2 such that its refined Humbert invariant q_C is equivalent to a given integral ternary quadratic form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14319 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constructing genus 2 curves with given refined Humbert invariants Kir, Harun Number Theory Algebraic Geometry In 1994, Kani introduced an algebraic version of the Humbert invariant, known as the refined Humbert invariant. This invariant q_C is a positive definite quadratic form attached to a smooth curve C of genus 2. It serves as a vital tool, as many geometric properties of C are reflected in the arithmetic properties of q_C. When the Jacobian J_C of a genus 2 curve C is isogenous to a product of an elliptic curve with complex multiplication, the forms q_C have been completely classified recently. In this paper, building upon this classification, we present a constructive algorithm that produces J_C and a divisorial representative of a curve C of genus 2 such that its refined Humbert invariant q_C is equivalent to a given integral ternary quadratic form. |
| title | Constructing genus 2 curves with given refined Humbert invariants |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2602.14319 |