A density result for real hyperelliptic curves

Brian Lawrence

doi:10.1016/j.crma.2016.10.014

Algebraic geometry

A density result for real hyperelliptic curves

Presented by: Jean-Pierre Serre

Brian Lawrence ¹

¹ Department of Mathematics, Building 380, Stanford University, Stanford, CA, 94305, USA

Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1219-1224.

is referenced by Corrigendum to: “A density result for real hyperelliptic curves” [C. R. Acad. Sci. Paris, Ser. I 354 (12) (2016) 1219–1224]

Abstract
Résumé

Let ${\infty^{+}, \infty^{-}}$ be the two points above ∞ on the real hyperelliptic curve $H : y^{2} = (x^{2} - 1) \prod_{i = 1}^{2 g} (x - a_{i})$ . We show that the divisor $([\infty^{+}] - [\infty^{-}])$ is torsion in Jac J for a dense set of $(a_{1}, a_{2}, \dots, a_{2 g}) \in {(- 1, 1)}^{2 g}$ . In fact, we prove by degeneration to a nodal $P^{1}$ that an associated period map has derivative generically of full rank.

Soient ${\infty^{+}, \infty^{-}}$ les deux points de la courbe hyperelliptique réelle $H : y^{2} = (x^{2} - 1) \prod_{i = 1}^{2 g} (x - a_{i})$ au-dessus du point ∞ de $P^{1}$ . On montre que le diviseur $([\infty^{+}] - [\infty^{-}])$ est de torsion dans Jac J pour un ensemble dense de $(a_{1}, a_{2}, \dots, a_{2 g}) \in {(- 1, 1)}^{2 g}$ . En fait, on démontre par réduction à un $P^{1}$ avec des points doubles que la dérivée d'un morphisme de périodes est génériquement surjectif.

Received: 2016-05-05
Accepted: 2016-10-14
Published online: 2016-11-09

DOI: 10.1016/j.crma.2016.10.014

Author's affiliations:

Brian Lawrence ¹

¹ Department of Mathematics, Building 380, Stanford University, Stanford, CA, 94305, USA

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Brian Lawrence. A density result for real hyperelliptic curves. Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1219-1224. doi : 10.1016/j.crma.2016.10.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.10.014/

References
Cited by

[1] A. Bogatyrev Extremal Polynomials and Riemann Surfaces, Springer, Berlin, 2012

[2] R.M. Robinson Conjugate algebraic integers in real point sets, Math. Z., Volume 84 (1964), pp. 415-427

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