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Generic models for genus 2 curves with real multiplication

Alex Cowan, Sam Frengley, Kimball Martin

Abstract

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental discriminants $D < 40$. A key step in our proof is to develop an algorithm for minimisation of conic bundles fibred over $\mathbb{P}^2$. We apply this algorithm to simplify the equations for the Mestre conic associated to the generic point on the Hilbert modular surface of fundamental discriminant $D < 100$ computed by Elkies--Kumar.

Generic models for genus 2 curves with real multiplication

Abstract

Explicit models of families of genus 2 curves with multiplication by are known for . We obtain generic models for genus 2 curves over with real multiplication in 12 new cases, including all fundamental discriminants . A key step in our proof is to develop an algorithm for minimisation of conic bundles fibred over . We apply this algorithm to simplify the equations for the Mestre conic associated to the generic point on the Hilbert modular surface of fundamental discriminant computed by Elkies--Kumar.
Paper Structure (25 sections, 12 theorems, 35 equations, 1 figure, 4 tables, 5 algorithms)

This paper contains 25 sections, 12 theorems, 35 equations, 1 figure, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Parametrising genus 2 curves with RM D
  3. An outline of the proof of Theorem 1.1: Simplifying the Mestre conic
  4. Further remarks
  5. Background
  6. Real multiplication and Hilbert modular surfaces
  7. The Mestre conic and cubic
  8. The Mestre conic associated to a Hilbert modular surface
  9. Simplified models for the generic Mestre conic
  10. Mestre conics for Igusa--Clebsch invariants
  11. Mestre conics for Elkies--Kumar models
  12. An algorithm for minimising a conic over "211A(t_1,t_2)
  13. Minimisation
  14. Searching for minimal models
  15. The main algorithm
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $D \in \{5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 44, 53, 61 \}$ and let $k$ be a field of characteristic $0$. Let $F_D(z, g, h, r, s; x) \in \mathbb Q(z,g,h,r,s)[x]$ be the sextic polynomial given in the electronic data associated to this paper electronic. Let $\mathscr{L}_D / \mathbb Q$ denot in $\mathbb{A}^5$, where $\lambda_D, q_D \in \mathbb Z[g,h]$ are the polynomials defined via eq:EK-

Figures (1)

  • Figure 1: For $D = 21$, the real values $(g,h)$ such that $\lambda_{21} = 0$ (red) and $q_{21} = 0$ (blue).

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2: A generic RM 12 family
  • Corollary 1.3: A generic RM 17 family
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.8
  • Remark 1.9
  • Remark 2.1
  • Proposition 2.2: Mestre mestre
  • ...and 21 more