Constructing genus 2 curves with given refined Humbert invariants

TL;DR

This paper develops a constructive algorithm to produce genus 2 curves with a specified refined Humbert invariant, linking geometric properties to arithmetic forms and extending recent classifications of Jacobians with complex multiplication.

Contribution

It introduces a new algorithm for constructing genus 2 curves with a given refined Humbert invariant based on recent classifications of their Jacobians.

Findings

Algorithm successfully constructs genus 2 curves with specified invariants

Extends classification of Jacobians with complex multiplication

Provides explicit divisorial representatives for curves

Abstract

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.