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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'' of a given genus ''g'', and their Jacobians ''J''. There is a moduli space ''M''''g'' of such curves, and a moduli space ''A''''g'' of abelian varieties of dimension ''g'', which are ''principally polarized''. There is a morphism :ι: ''M''''g'' → ''A''''g'' 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 ''M''''g'' is 3''g'' − ''3'', for ''g'' ≥ 2, while the dimension of ''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 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 ''A''''g''. 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schottky problem」の詳細全文を読む スポンサード リンク
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