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theta characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve ''C'' is a divisor class Θ such that 2Θ is the canonical class, In terms of holomorphic line bundles ''L'' on a connected compact Riemann surface, it is therefore ''L'' such that ''L''2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by
==History and genus 1==
The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve ''E'' over the complex numbers are seen to be in 1-1 correspondence with the four points ''P'' on ''E'' with 2''P'' = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when ''E'' is treated as a complex torus.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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