[Paper Review] Real Multiplication on Jacobian Varieties

Robert A. Kucharczyk|arXiv (Cornell University)|Jan 7, 2012

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper investigates real multiplication on Jacobian varieties of Wiman curves $W_g$, showing that for certain abelian differentials $(W_g, \omega_k)$, the Jacobian admits real multiplication by the trace field of the Veech group. While this structure holds for minimal and maximal $k$ with $\gcd(k, 2g+1) = t$, it fails to be preserved under the $\mathrm{SL}_2(\mathbb{R})$-action for intermediate $k$, providing counterexamples to a proposed extension of Möller’s theorem beyond Veech surfaces.

ABSTRACT

This is a slightly revised version of the author's 2010 diploma thesis. It is concerned with the interplay between real multiplication on Jacobian varieties, as the title suggests, and complex geodesics in the moduli space of curves. Large parts are expository and may hopefully serve as a very incomplete introduction to Teichmueller disks and curves, the moduli space of abelian differentials with its SL2(R)-operation and variations of Hodge structure.

Motivation & Objective

• To understand the interplay between real multiplication on Jacobian varieties and the $\mathrm{SL}_2(\mathbb{R})$-action on Teichmüller disks in the moduli space of curves.
• To determine whether real multiplication structures on Jacobians of non-Veech translation surfaces are preserved under the $\mathrm{SL}_2(\mathbb{R})$-action.
• To analyze the variation of Hodge structures over Teichmüller disks and identify conditions under which real multiplication arises.
• To provide explicit counterexamples to a generalized version of Möller’s theorem on real multiplication in the absence of Veech surface structure.

Proposed method

• The study employs the $\mathrm{SL}_2(\mathbb{R})$-action on the moduli space of abelian differentials to analyze Teichmüller disks associated with translation surfaces $(W_g, \omega_k)$.
• It uses the Veech group and its trace field to define a candidate real multiplication structure on the Jacobian of $W_g$.
• The Ahlfors-Rauch variational formula is applied to compute the derivative of the period matrix along the Teichmüller disk, detecting non-invariance of the Hodge structure.
• A symplectic basis of homology is constructed to express the period matrix, enabling explicit computation of Hodge structure components.
• The paper analyzes the behavior of the period matrix under the $\mathrm{SL}_2(\mathbb{R})$-action, particularly at $\tau = i$, to detect failure of invariance.
• It compares the structure of the canonical subspace and eigenforms under the action, showing that real multiplication is not preserved for intermediate $k$.

Experimental results

Research questions

• RQ1 Does real multiplication on the Jacobian of a non-Veech translation surface persist under the $\mathrm{SL}_2(\mathbb{R})$-action on the moduli space of abelian differentials?
• RQ2 For which values of $k$ in the family $(W_g, \omega_k)$ does the real multiplication structure on $J(W_g)$ remain invariant under the $\mathrm{SL}_2(\mathbb{R})$-action?
• RQ3 Can the trace field of the Veech group of a translation surface be realized as a real multiplication on the Jacobian, even when the surface is not a Veech surface?
• RQ4 How does the variation of Hodge structure over a Teichmüller disk relate to the existence and invariance of real multiplication?
• RQ5 To what extent does the failure of $\mathrm{SL}_2(\mathbb{R})$-invariance of real multiplication structures provide counterexamples to a generalized Möller’s theorem?

Key findings

• For every $g \geq 3$ and $1 < k < g$, the Jacobian $J(W_g)$ admits real multiplication by the trace field of $\mathrm{SL}(W_g, \omega_k)$, with $\omega_k$ as an eigenform for the identity character.
• The real multiplication structure is preserved under the $\mathrm{SL}_2(\mathbb{R})$-action if and only if $k = t$ or $k = g$ where $t = \gcd(k, 2g+1)$, i.e., for minimal and maximal $k$ in each $\gcd$-class.
• For intermediate $k$ with $\gcd(k, 2g+1) = t$, the real multiplication structure is not preserved by the $\mathrm{SL}_2(\mathbb{R})$-action, as shown by non-vanishing derivative of the period matrix component $\Pi_{k-\ell t, k+\ell t}$ at $\tau = i$.
• The derivative computation via the Ahlfors-Rauch formula yields $\left| \frac{d\Pi_{k-\ell t,k+\ell t}}{d\tau} \right|_{\tau=i} = \int_X |\omega_k|^2 > 0$, contradicting invariance.
• This provides explicit counterexamples to an extension of Möller’s theorem to non-Veech surfaces, valid for all $g \geq 3$.
• The construction generalizes McMullen’s $g=3, k=2$ case and corrects an erroneous claim in Möller’s work by showing that not all $1 < k < g$ yield counterexamples, only those with $\gcd(k, 2g+1) = t$ and $k \neq t, g$.