Abstract
To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. We show that in the case of a hyperelliptic surface this geometric invariant is bounded from above by a constant and that for any surface of genus g the square of the second successive minimum is equally of order log(g). We obtain improved bounds for the kth successive minimum of the Jacobian, if the surface contains small simple closed geodesics.
Similar content being viewed by others
References
Bavard, C.: La systole des surfaces hyperelliptiques, vol. 71. Prepublication de l’ENS Lyon (1992)
Bergé A.-M., Martinet J.: Densité dans des familles de réseaux. Application aux réseaux isoduaux. L’enseignement Mathématique 41, 335–365 (1995)
Birkenhake C., Lange H.: Complex Abelian Varieties, Grundlehren der mathematischen Wissenschaften, vol. 302. Springer, Berlin (2004)
Buser P.: Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, vol. 106. Birkhäuser Verlag, Boston (1992)
Buser P., Sarnak P.: On the period matrix of a Riemann surface of large genus (with an appendix by Conway, J. H. and Sloane, N. J. A.). Inventiones Mathematicae 117(1), 27–56 (1994)
Fay J.D.: Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, vol. 352. Springer, Berlin (1973)
Gendulphe M.: Découpages et inégalités systoliques pour les surfaces hyperboliques à à bord. Geometriae dedicata 142, 23–35 (2009)
Gruber P.M., Lekkerkerker C.G.: Geometry of Numbers, North Holland Mathematical Library. Elsevier, Amsterdam (1987)
Jenni F.: The 1st eigenvalue of Laplace operators in selected examples of Riemannian surfaces. Commentarii Mathematici Helvetici 59(2), 193–203 (1984)
Parlier, H.: On the geometry of simple closed geodesics. PhD thesis, Ecole Polytechnique Fédérale de Lausanne (2004)
Parlier H.: Lengths of geodesics on Riemann surfaces with boundary. Ann. Acad. Sci. Fenn. Math. 30(2), 227–236 (2005)
Schmutz P.: Riemann surfaces with shortest geodesic of maximal length. Geometric and Functional Analysis 3(6), 564–631 (1993)
Schottky F.: Zur Theorie der Abelschen Functionen von vier Variablen. Journal für die reine und angewandte Mathematik 102, 304–352 (1888)
Shiota T.: Characterization of Jacobian varieties in terms of soliton-equations. Inventiones Mathematicae 83(2), 333–382 (1986)
van Geemen B.: Siegel modular forms vanishing on the moduli space of curves. Inventiones Mathematicae 78(2), 329–349 (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author has been supported by the Swiss National Science Foundation (grant nr. 200021-125324).
Rights and permissions
About this article
Cite this article
Muetzel, B. On the second successive minimum of the Jacobian of a Riemann surface. Geom Dedicata 161, 85–107 (2012). https://doi.org/10.1007/s10711-012-9695-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-012-9695-3