[Paper Review] Shimura curves within the locus of hyperelliptic Jacobians in genus three

Samuel Grushevsky, Martin Moeller|arXiv (Cornell University)|Aug 23, 2013

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper constructs infinitely many Shimura curves within the locus of hyperelliptic Jacobians in genus 3, using explicit period matrix parametrizations and degeneration data analysis. It proves that any non-complete Kuga curve in this locus must degenerate with equal growth orders in a specific $σ_{1+1}$ type, and shows these curves are not complete in the moduli space of abelian threefolds.

ABSTRACT

We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained within a divisor in the moduli space of abelian varieties.

Motivation & Objective

• To investigate the existence and structure of Shimura and Kuga curves within the locus of hyperelliptic Jacobians in genus 3.
• To determine the necessary degeneration types for non-complete Kuga curves in the hyperelliptic locus of $\mathcal{A}_3$.
• To construct explicit families of Shimura curves in the hyperelliptic locus using period matrices with complex parameters.
• To demonstrate that infinitely many distinct Shimura curves exist in the hyperelliptic locus of genus 3 abelian varieties.
• To develop general techniques for studying Shimura curves inside divisors in the moduli space of principally polarized abelian varieties.

Proposed method

• Uses period matrices of the form $\Pi_u(t) = \begin{pmatrix} t + iu^2 & u^2/2 & iu \\ u^2/2 & t & u \\ iu & u & i \end{pmatrix}$ with $t \in \mathbb{H}$ and $u \in \mathbb{Q}+i\mathbb{Q} \setminus \mathbb{Z}+i\mathbb{Z}$ to parametrize families of abelian threefolds.
• Applies base changes via symplectic matrices $B_D$, $B_H$, $S$, and $S_3$ to diagonalize monodromy actions on homology and differential forms.
• Employs toroidal compactifications $\overline{\mathcal{A}_g}$ to analyze degeneration behavior of modular forms and period matrices.
• Uses isogenies between Jacobians and product abelian varieties, represented by matrices like $S_3 D (B_H^{-1})^T$, to relate period matrices across isogeny classes.
• Analyzes the quotient family $Y_s$ of genus 2 curves via the involution $T_0T^3$, using known period matrices from [BL04] for Type II curves.
• Composes symplectic matrices $C_1$ and $C_2$ to transform the period matrix into the required form for verifying Shimura curve structure.

Experimental results

Research questions

• RQ1 What degeneration types are possible for non-complete Kuga curves generically contained in the hyperelliptic locus of genus 3 Jacobians?
• RQ2 Can infinitely many distinct Shimura curves be explicitly constructed within the hyperelliptic locus of $\mathcal{A}_3$?
• RQ3 How do period matrices of such curves behave under symplectic base changes and monodromy actions?
• RQ4 What is the role of complex multiplication and CM points in the construction of these Shimura curves?
• RQ5 To what extent do these curves persist under deformation, and how do isogenies relate their period matrices?

Key findings

• Any non-complete Kuga curve in $\mathcal{A}_3$ generically contained in the hyperelliptic locus must degenerate with equal growth orders in a $\sigma_{1+1}$ type, as specified by a period matrix of the form $\begin{pmatrix} t & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 0 \end{pmatrix} + M$.
• For any $u \in \mathbb{Q}+i\mathbb{Q} \setminus \mathbb{Z}+i\mathbb{Z}$, the family $\Pi_u(t)$ defines a Shimura curve whose generic point lies in the hyperelliptic locus $\mathrm{Hyp}_3$.
• There exist infinitely many distinct Shimura curves in $\mathrm{Hyp}_3$, as proven in Proposition 7.4.
• The Shimura curve for $u = \frac{1+i}{2}$ is isomorphic to a known example in the Moonen-Oort list and corresponds to hyperelliptic curves with reduced automorphism group $\mathbb{Z}_2 \times \mathbb{Z}_4$.
• The period matrix of the family $X_s$ is transformed via symplectic matrices to match the form required by Lemma 4.3, confirming the Shimura curve structure.
• The isogeny from the Jacobian of $X_{1/2}$ to a product with period matrix $\begin{pmatrix} Z_2 & 0 \\ 0 & i \end{pmatrix}$ is given by $S_3 D (B_H^{-1})^T$, with $D = \mathrm{diag}(1,1,1,2,1,1)$.