Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-09T12:41:06.388Z Has data issue: false hasContentIssue false
  • English
  • Français

The non-existence of stable Schottky forms

Part of: Curves Abelian varieties and schemes Deformations of analytic structures

Published online by Cambridge University Press:  10 March 2014

G. Codogni  and
N. I. Shepherd-Barron
G. Codogni
Affiliation:
DPMMS, University of Cambridge, Cambridge CB3 0WB, UK email g.codogni@dpmms.cam.ac.uk
N. I. Shepherd-Barron
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK email nisb@dpmms.cam.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We show that there is no stable Siegel modular form that vanishes on every moduli space of curves.

Keywords

curveabelian varietymoduliSchottky problem

MSC classification

Secondary: 14H40: Jacobians, Prym varieties 14H42: Theta functions; Schottky problem 14K25: Theta functions 32G20: Period matrices, variation of Hodge structure; degenerations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014 , pp. 679 - 690
Copyright
© The Author(s) 2014 

References

Alexeev, V. A. and Brunyate, A., Extending the Torelli map to toroidal compactifications of Siegel space, Invent. Math. 188 (2012), 175–196.CrossRefGoogle Scholar
Codogni, G., Non-perturbative Schottky problem and stable equations for the hyperelliptic locus, Preprint (2013), arXiv:1306.1183 [math.AG].Google Scholar
Fay, J., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, vol. 352 (Springer, 1973).Google Scholar
Freitag, E., Stabile Modulformen, Math. Ann. 230 (1977), 197–211.Google Scholar
Freitag, E., Siegelsche Modulfunktionen (Springer, 1983).Google Scholar
Grushevsky, S. and Salvati Manni, R., The superstring cosmological constant and the Schottky form in genus 5, Amer. J. Math. 133 (2011), 1007–1037.Google Scholar
Hoyt, W., On products and algebraic families of Jacobian varieties, Ann. of Math. (2) 77 (1963), 414–423.Google Scholar
Igusa, J.-I., Schottky’s invariant and quadratic forms, in E. B. Christoffel: the influence of his work on mathematics and the physical sciences (Birkhäuser, Basel, 1981); 352–362.Google Scholar
Igusa, J.-I., On the irreducibility of Schottky’s divisor, J. Fac. Sci. Univ. Tokyo Section IA Math. 28 (1981), 531–545.Google Scholar
Mumford, D., Further comments on boundary points, inProc. Woods Hole conference, Woods Hole, MA, 1964.Google Scholar
Schottky, F., Zur Theorie der Abelschen Functionen von vier Variabeln, J. Reine Angew. Math. 102 (1888), 304–352.Google Scholar
Shepherd-Barron, N. I., Siegel modular forms and the gonality of curves, Preprint (2013),arXiv:1306.6253 [math.AG].Google Scholar
Springer, G., Introduction to Riemann surfaces (Addison-Wesley, 1957).Google Scholar
Yamada, A., Precise variational formulas for abelian differentials, Kodai Math. J. 3 (1980), 114–143.Google Scholar