Shimura subvarieties in the Prym locus of ramified Galois coverings

• Grosselli, Gian Paolo ^[1] ; Mohajer, Abolfazl ^[2]
  1. [1] Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100, Pavia, Italy
  2. [2] Institut für Mathematik, Fachbereich 08, Universität Mainz, 55099, Mainz, Germany
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 1, 2023, págs. 199-218
• Idioma: inglés
• DOI: 10.1007/s13348-021-00342-5
• Enlaces
  • Texto completo
• Resumen

  • We study Shimura (special) subvarieties in the moduli space A_{p,D} of complex abelian varieties of dimension p and polarization type D. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to {{\mathbb {P}}}^1. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

• Referencias bibliográficas
  • Birkenhake, C., Lange, H.: Complex abelian varieties. In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical...
  • Colombo, E., Frediani, P.: A bound on the dimension of a totally geodesic submanifold in the Prym locus. Collect. Math. 70(1), 51–57 (2019)....
  • Colombo, E., Frediani, P.: On the dimension of totally geodesic submanifolds in the Prym loci. Boll Unione Mat. Ital. (2021). https://doi-org.sire.ub.edu/10.1007/s40574-021-00287-4....
  • Colombo, E., Frediani, P., Ghigi, A., Penegini, M.: Shimura curves in the Prym locus. In: Communications in Contemporary Mathematics, vol....
  • Grosselli, G.P., Frediani, P.: Shimura curves in the Prym loci of ramified double covers. arXiv:2007.09646
  • Frediani, P., Ghigi, A., Penegini, M.: Shimura varieties in the Torelli locus via Galois coverings. Int. Math. Res. Not. IMRN 20, 10595–10623...
  • Frediani, P., Grosselli, G.P., Mohajer, A.: Higher dimensional Shimura varieties in the Prym loci of ramified double covers. arXiv:2101.09016
  • Frediani, P., Ghigi, A., Spelta, I.: Infinitely many Shimura varieties in the Jacobian locus for g \le 4. Ann. Scuola Norm. Sup. Pisa Cl....
  • Mohajer, A.: On the Prym map of Galois coverings. Rocky Mt. J. Math. arXiv:2004.09678
  • Mohajer, A., Zuo, K.: On Shimura subvarieties generated by families of abelian covers of \mathbb{P}^{1}. J. Pure Appl. Algebra 222(4), 931–949...
  • Moonen, B.: Special subavarieties arising from families of cyclic covers of the projective line. Doc. Math. 15, 793–819 (2010)
  • Moonen, B., Oort, F.: The Torelli locus and special subvarieties. In: Handbook of Moduli, vol. II, International Press, Boston, pp. 549–94...
  • Pardini, R.: Abelian covers of algebraic varieties. J. R. Angew. Math. 417, 191–213 (1991)
  • Recillas, S., Rodríguez, R.: Prym varieties and fourfold covers. Publ. Preliminares Inst. Mat. Univ. Nac. Aut. Mexico (2003). arXiv:math/0303155
  • Szamuely, T.: Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, vol. 117. Cambridge University Press, Cambridge...
  • Völklein, H.: Groups as Galois Groups: An Introduction. Cambridge Studies in Advanced Mathematics, vol. 53. Cambridge University Press, Cambridge...
  • Wright, A.: Shwarz triangle mappings and Teichmüller curves: abelian square-tiled surfaces. J. Mod. Dyn. 6, 405–426 (2012)