Problem 4.12 — (Distortion of the Schottky locus).

Definitions and mathematical statement

Let $\mathcal A_g$ be the moduli space of principally polarized complex abelian varieties of dimension $g$, and let $j:\mathcal M_g\to\mathcal A_g$ be the Jacobian (Torelli) map. The Schottky or Jacobian locus is customarily $\mathcal J_g=j(\mathcal M_g)$; some conventions instead use an appropriate closure, which must be distinguished.

Problem 4.12 asks to compute the distortion of the Schottky locus in $\mathcal A_g$. The excerpt says only that a naive guess might be exponential; it does not state exponential distortion as an established fact or a fully specified conjecture.

To make distortion precise one must fix a metric on $\mathcal A_g$, a metric on $\mathcal J_g$ (for example an intrinsic path metric, or one transferred from a metric on curve moduli), and a comparison convention. Under one possible all-pairs convention the distortion function is \[ \Delta(R)=\sup\{d_{\mathcal J}(x,y):x,y\in\mathcal J_g, d_{\mathcal A}(x,y)\leq R\}. \] The problem would then ask for its growth, allowing that the supremum might be infinite. This formula is illustrative only: the excerpt supplies no metrics, basepoint/uniformity choice, or coarse equivalence relation for growth functions. In particular, the Siegel metric on $\mathcal A_g$ and either Teichmüller or intrinsic induced metrics on the source cannot be selected silently. The genus range and image-versus-closure convention also matter.

Short English statement

Determine how distances within the Jacobian locus compare with ambient distances in abelian-variety moduli. The source tentatively suggests exponential distortion, but omits the metrics and growth convention.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000049$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 4.12, PDF page 42: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; page-disambiguated numbering among 329 headings.
• No additional authoritative quantitative-distortion reference is supplied. The registry explicitly says the relevant quantitative literature could not be verified.