## Revista Matemática Iberoamericana

### Group actions on Jacobian varieties

Anita M. Rojas

#### Abstract

Consider a finite group $G$ acting on a Riemann surface $S$, and the associated branched Galois cover $\pi_G:S \to Y=S/G$. We introduce the concept of \emph{geometric signature} for the action of $G$, and we show that it captures much information: the geometric structure of the lattice of intermediate covers, the isotypical decomposition of the rational representation of the group $G$ acting on the Jacobian variety $JS$ of $S$, and the dimension of the subvarieties of the isogeny decomposition of $JS$. We also give a version of Riemann's existence theorem, adjusted to the present setting.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 23, Number 2 (2007), 397-420.

Dates
First available in Project Euclid: 26 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1190831216

Mathematical Reviews number (MathSciNet)
MR2371432

Zentralblatt MATH identifier
1139.14026

#### Citation

Rojas, Anita M. Group actions on Jacobian varieties. Rev. Mat. Iberoamericana 23 (2007), no. 2, 397--420. https://projecteuclid.org/euclid.rmi/1190831216

#### References

• Breuer, T.: Characters and Automorphism Groups of Compact Riemann Surfaces. London Mathematical Society Lecture Note Series 280. Cambridge University Press, 2000.
• Broughton, S. A.: Classifying finite group actions on surfaces of low genus. J. Pure Appl. Algebra 69 (1991), no. 3, 233-270.
• Carocca, A. and Rodríguez, R.: Jacobians with group actions and rational idempotents. J. Algebra 306 (2006), no. 2, 322-343.
• Curtis, C. and Reiner, I.: Representation theory of finite groups and associative algebras. Pure and applied mathematics XI. Interscience Publishers, New-York-London, 1962.
• Farkas, H. and Kra, I.: Riemann Surfaces. Graduate Texts in Mathematics 71. Springer-Verlag, New York, 1991.
• G.A.P. Groups, Algorithm and Programming Computer Algebra System. http://www.gap-system.org/$\sim$gap.
• Jones, G. A. and Singerman, D.: Complex functions: An algebraic and geometric viewpoint. Cambridge University Press, Cambridge, 1987.
• Ksir, A.: Dimensions of Prym Varieties. Int. J. Math. Math. Sci. 26 (2001), no. 2, 107-116.
• Lange, H. and Recillas, S.: Abelian Varieties with group action. J. Reine Angew. Math. 575 (2004), 135-155.
• Miranda, R.: Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics 5. American Mathematical Society, Providence, 1995.
• Recillas, S. and Rodríguez, R.: Jacobians and Representations for $S_3$. In Workshop on Abelian Varieties and Theta Functions (Morelia, 1996), 117-140. Aportaciones Mat. Investig. 13. Soc. Mat. Mexicana, México, 1998.
• Rojas, A. M.: Group actions on Jacobian varieties. Ph.D. Thesis. Pontificia Universidad Católica de Chile, 2002.
• Sánchez-Argáez, A.: Acciones de $A_5$ en Jacobianas de curvas. Aportaciones Mat., Comun. 25 (1999), 99-108.
• Serre, J. P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics 42. Springer, 1996.
• Singerman, D.: Subgroups of Fuchsian groups and finite permutation groups. Bull. London Math. Soc. 2 (1972), 319-323.
• Streit, M.: Period Matrices and Representation Theory. Abh. Math. Sem. Univ. Hamburg 71 (2001), 279-290.
• Suzuki, M.: Group Theory I. Grundlehren der Mathematischen Wissenschaften 247. Springer-Verlag, Berlin-New York, 1982.
• Völklein, H.: Groups as Galois groups. Cambridge studies in Advanced Mathematics 53. Cambridge University Press, 1996.
• Wolfart, J.: Regular dessins, endomorphisms of Jacobians, and transcendence. In A Panorama of Number Theory or The View from Baker's Garden (Zürich, 1999), 107-120. Cambridge University Press, Cambridge, 2002.
• Wolfart, J.: Triangle groups and Jacobians of CM type. Manuscript, Frankfurt a.M., 2000. http://www.math.uni-frankfurt.de/$\sim$wolfart.