Constructing genus 2 curves with given refined Humbert invariants
Abstract
In 1994, Kani introduced an algebraic version of the Humbert invariant, known as the refined Humbert invariant. This invariant qC 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 qC. When the Jacobian JC of a genus 2 curve C is isogenous to a product of an elliptic curve with complex multiplication, the forms qC have been completely classified recently. In this paper, building upon this classification, we present a constructive algorithm that produces JC and a divisorial representative of a curve C of genus 2 such that its refined Humbert invariant qC is equivalent to a given integral ternary quadratic form.
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