Geometric Formulation
More precisely, one should consider algebraic curves C of a given genus g, and their Jacobians J. There is a moduli space Mg of such curves, and a moduli space Ag of abelian varieties of dimension g, which are principally polarized. There is a morphism
- ι: Mg → Ag
which on points (geometric points, to be more accurate) takes C to J. The content of Torelli's theorem is that ι is injective (again, on points). The Schottky problem asks for a description of the image of ι.
It is discussed for g ≥ 4: the dimension of Mg is 3g − 3, for g ≥ 2, while the dimension of Ag 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 is the first interesting case, 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 Ag. A more precise form of the question is to determine whether the image of ι essentially coincides with the Schottky locus (in other words, whether it is Zariski dense there).
Read more about this topic: Schottky Problem
Famous quotes containing the words geometric and/or formulation:
“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o’ th’ day
The clock doth strike, by algebra.”
—Samuel Butler (1612–1680)
“You do not mean by mystery what a Catholic does. You mean an interesting uncertainty: the uncertainty ceasing interest ceases also.... But a Catholic by mystery means an incomprehensible certainty: without certainty, without formulation there is no interest;... the clearer the formulation the greater the interest.”
—Gerard Manley Hopkins (1844–1889)