BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:COMPUTATIONAL APPROACH TO THE SCHOTTKY PROBLEM - Christian Klein (
 Université de Bourgogne)
DTSTART:20230725T123000Z
DTEND:20230725T133000Z
UID:TALK202831@talks.cam.ac.uk
DESCRIPTION:We present a computational approach to the classical Schottky 
 problem based on Fay&rsquo\;s trisecant identity for genus&nbsp\;g&nbsp\;&
 ge\;&nbsp\;4. For a given Riemann matrix&nbsp\;B&nbsp\;&isin\;&nbsp\;Hg\, 
 the Fay identity establishes linear dependence of secants in the Kummer va
 riety if and only if the Riemann matrix corresponds to a Jacobian variety 
 as shown by Krichever. The theta functions in terms of which these secants
  are expressed depend on the Abel maps of four arbitrary points on a Riema
 nn surface. However\, there is no concept of an Abel map for general&nbsp\
 ;B&nbsp\;&isin\;&nbsp\;Hg. To establish linear dependence of the secants\,
  four components of the vectors entering the theta functions can be chosen
  freely. The remaining components are determined by a Newton iteration to 
 minimize the residual of the Fay identity. Krichever&rsquo\;s theorem assu
 res that if this residual vanishes within the finite numerical precision f
 or a generic choice of input data\, then the Riemann matrix is with this n
 umerical precision the period matrix of a Riemann surface. The algorithm i
 s compared in genus 4 for some examples to the Schottky-Igusa modular form
 \, known to give the Jacobi locus in this case. It is shown that the same 
 residuals are achieved by the Schottky-Igusa form and the approach based o
 n the Fay identity in this case. In genera 5\, 6 and 7\, we discuss known 
 examples of Riemann matrices and perturbations thereof for which the Fay i
 dentity is not satisfied. This is work with E. Brandon de Leon and J. Frau
 endiener.
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR