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.
Geometric formulation More precisely, one should consider algebraic curves C {\displaystyle C} of a given genus g {\displaystyle g} , and their Jacobians Jac ( C ) {\displaystyle \operatorname {Jac} (C)} . There is a moduli space M g {\displaystyle {\mathcal {M}}_{g}} of such curves, and a moduli space of abelian varieties, A g {\displaystyle {\mathcal {A}}_{g}} , of dimension g {\displaystyle g} , which are principally polarized. There is a morphism Jac : M g → A g {\displaystyle \operatorname {Jac} :{\mathcal {M}}_{g}\to {\mathcal {A}}_{g}} which on points (geometric points, to be more accurate) takes isomorphism class [ C ] {\displaystyle [C]} to [ Jac ( C ) ] {\displaystyle [\operatorname {Jac} (C)]} . The content of Torelli's theorem is that Jac {\displaystyle \operatorname {Jac} } is injective (again, on points). The Schottky problem asks for a description of the image of Jac {\displaystyle \operatorname {Jac} } , denoted J g = Jac ( M g ) {\displaystyle {\mathcal {J}}_{g}=\operatorname {Jac} ({\mathcal {M}}_{g})} . The dimension of M g {\displaystyle {\mathcal {M}}_{g}} is 3 g − 3 {\displaystyle 3g-3} , for g ≥ 2 {\displaystyle g\geq 2} , while the dimension of A g {\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 g = 4 {\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 A g {\displaystyle {\mathcal {A}}_{g}} . A more precise form of the question is to determine whether the image of Jac {\displaystyle \operatorname {Jac} } essentially coincides with the Schottky locus (in other words, whether it is Zariski dense there).
Dimension 1 case All elliptic curves are the Jacobian of themselves, hence the moduli stack of elliptic curves M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} is a model for A 1 {\displaystyle {\mathcal {A}}_{1}} .
Dimensions 2 and 3 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 M 2 , M 1 , 1 × M 1 , 1 {\displaystyle {\mathcal {M}}_{2},{\mathcal {M}}_{1,1}\times {\mathcal {M}}_{1,1}} embed into A 2 {\displaystyle {\mathcal {A}}_{2}} . There is a similar description for dimension 3 since an Abelian variety can be the product of Jacobians.
Period lattice formulation If one describes the moduli space A g {\displaystyle {\mathcal {A}}_{g}} in intuitive terms, as the parameters on which an abelian variety depends, then the Schottky problem asks simply what condition on the parameters implies that the abelian variety comes from a curve's Jacobian. The classical case, over the complex number field, has received most of the attention, and then an abelian variety A is simply a complex torus of a particular type, arising from a lattice in Cg. In relatively concrete terms, it is being asked which lattices are the period lattices of compact Riemann surfaces.
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