---
tag: C210P5AK7X
kind: problem
name: Schottky problem
msc: [14H42, 14K10]
uses: [R7W03TXRS9, ZQ84M9D9TC, BRKFX8BQ60, 2F5CYQJVAF]
about: [ZQ84M9D9TC, R7W03TXRS9]
sources:
  - url: https://en.wikipedia.org/wiki/Schottky_problem
same_as:
  wikidata: Q7432779
  wikipedia: Schottky problem
history:
  posed_by: "Friedrich Schottky"
  first_stated: "1888"
---

## Problem

Let $g \ge 1$, and let $\mathcal{A}_g$ be the moduli space of principally polarized complex [[abelian varieties|R7W03TXRS9]] of dimension $g$.
The [[Jacobian variety|ZQ84M9D9TC]] of a smooth projective complex curve of genus $g$, with its theta divisor, is a point of $\mathcal{A}_g$.
Let $\mathcal{J}_g \subseteq \mathcal{A}_g$ be the set of these points.
Characterize the points of $\mathcal{J}_g$, or of its closure, among all points of $\mathcal{A}_g$.

## Remarks

By the [[Torelli theorem|BRKFX8BQ60]], a curve is determined by its Jacobian with its polarization, so $\mathcal{J}_g$ has the dimension $3g - 3$ of the moduli of curves when $g \ge 2$.
The space $\mathcal{A}_g$ has dimension $g(g + 1)/2$.
For $g \le 3$ the closure of $\mathcal{J}_g$ is all of $\mathcal{A}_g$, and for $g \ge 4$ it is a proper closed subset.
For $g = 4$, Schottky found a modular form, built from [[theta functions|2F5CYQJVAF]], that vanishes on $\mathcal{J}_4$.
Igusa showed that its zero locus is exactly the closure of $\mathcal{J}_4$.
In 1986 Shiota proved a conjecture of Novikov on this problem.
It says that an indecomposable principally polarized abelian variety is a Jacobian if and only if its theta function solves the KP equation.
Other characterizations, for example by trisecants of the Kummer variety, are also known.

## Explanations

### L1 human

Every curve has an abelian variety attached to it, its Jacobian.
There are many more abelian varieties than curves once the dimension is at least $4$.
The Schottky problem asks how to recognize, from its equations or its theta function, which abelian varieties come from curves.