Period matrices and the Schottky problem. (English) Zbl 1028.14009
Bujalance, E. (ed.) et al., Topics on Riemann surfaces and Fuchsian groups. Based on lectures from the conference, Madrid, Spain, 1998. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 287, 155-163 (2001).
Introduction: In this talk we explain what is the period matrix of an algebraic curve (over \(\mathbb{C})\) and prove some of its properties (Riemann’s equality and Riemann’s inequality). We explain how are related period matrices of isomorphic curves and describe the action of the symplectic group on the space of period matrices. We will state (without proof) the converse of this construction which is Torelli’s theorem and develop some of the consequences of this theorem. We will then state Schottky’s problem, and explain how one can check, using Theta characteristics, if a given matrix is the period matrix of a smooth genus 2 or 3 curve, and finally state the solution to Schottky’s problem in genus 4.
For the entire collection see [Zbl 0964.00037].
For the entire collection see [Zbl 0964.00037].
MSC:
| 14H42 | Theta functions and curves; Schottky problem |
| 30F10 | Compact Riemann surfaces and uniformization |
| 32G20 | Period matrices, variation of Hodge structure; degenerations |
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
