In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.

More precisely, one should consider algebraic curves ${\displaystyle C}$ of a given genus ${\displaystyle g}$, and their Jacobians ${\displaystyle \operatorname {Jac} (C)}$. There is a moduli space ${\displaystyle {\mathcal {M}}_{g}}$ of such curves, and a moduli space of abelian varieties, ${\displaystyle {\mathcal {A}}_{g}}$, of dimension ${\displaystyle g}$, which are principally polarized. There is a morphism

${\displaystyle \operatorname {Jac} :{\mathcal {M}}_{g}\to {\mathcal {A}}_{g}}$

which on points (geometric points, to be more accurate) takes isomorphism class ${\displaystyle [C]}$ to ${\displaystyle [\operatorname {Jac} (C)]}$. The content of Torelli's theorem is that ${\displaystyle \operatorname {Jac} }$ is injective (again, on points). The Schottky problem asks for a description of the image of ${\displaystyle \operatorname {Jac} }$, denoted ${\displaystyle {\mathcal {J}}_{g}=\operatorname {Jac} ({\mathcal {M}}_{g})}$.

The dimension of ${\displaystyle {\mathcal {M}}_{g}}$ is ${\displaystyle 3g-3}$, for ${\displaystyle g\geq 2}$, while the dimension of ${\displaystyle {\mathcal {A}}_{g}}$ is g(g + 1)/2. This means that the dimensions are the same (0, 1, 3, 6) for g = 0, 1, 2, 3. Therefore ${\displaystyle g=4}$ is the first case where the dimensions change, and this was studied by F. Schottky in the 1880s. Schottky applied the theta constants, which are modular forms for the Siegel upper half-space, to define the Schottky locus in ${\displaystyle {\mathcal {A}}_{g}}$. A more precise form of the question is to determine whether the image of ${\displaystyle \operatorname {Jac} }$ essentially coincides with the Schottky locus (in other words, whether it is Zariski dense there).

All elliptic curves are the Jacobian of themselves, hence the moduli stack of elliptic curves ${\displaystyle {\mathcal {M}}_{1,1}}$ is a model for ${\displaystyle {\mathcal {A}}_{1}}$.

In the case of Abelian surfaces, there are two types of Abelian varieties: the Jacobian of a genus 2 curve, or the product of Jacobians of elliptic curves. This means the moduli spaces

${\displaystyle {\mathcal {M}}_{2},{\mathcal {M}}_{1,1}\times {\mathcal {M}}_{1,1}}$