Involve: A Journal of Mathematics

  • Involve
  • Volume 9, Number 4 (2016), 639-655.

Jacobian varieties of Hurwitz curves with automorphism group $\mathrm{PSL}(2,q)$

Allison Fischer, Mouchen Liu, and Jennifer Paulhus

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Abstract

The size of the automorphism group of a compact Riemann surface of genus g > 1 is bounded by 84(g 1). Curves with automorphism group of size equal to this bound are called Hurwitz curves. In many cases the automorphism group of these curves is the projective special linear group PSL(2,q). We present a decomposition of the Jacobian varieties for all curves of this type and prove that no such Jacobian variety is simple.

Article information

Source
Involve, Volume 9, Number 4 (2016), 639-655.

Dates
Received: 5 February 2015
Revised: 8 July 2015
Accepted: 20 July 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511371048

Digital Object Identifier
doi:10.2140/involve.2016.9.639

Mathematical Reviews number (MathSciNet)
MR3530204

Zentralblatt MATH identifier
1345.14038

Subjects
Primary: 14H40: Jacobians, Prym varieties [See also 32G20] 14H37: Automorphisms 20G05: Representation theory

Keywords
Jacobian varieties Hurwitz curves projective special linear group representation theory

Citation

Fischer, Allison; Liu, Mouchen; Paulhus, Jennifer. Jacobian varieties of Hurwitz curves with automorphism group $\mathrm{PSL}(2,q)$. Involve 9 (2016), no. 4, 639--655. doi:10.2140/involve.2016.9.639. https://projecteuclid.org/euclid.involve/1511371048


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References

  • G. Cardona, “$\mathbb Q$-curves and abelian varieties of $\rm GL\sb 2$-type from dihedral genus 2 curves”, pp. 45–52 in Modular curves and abelian varieties, edited by J. Cremona et al., Progr. Math. 224, Birkhäuser, Basel, 2004.
  • M. Conder, “Hurwitz groups: a brief survey”, Bull. Amer. Math. Soc. $($N.S.$)$ 23:2 (1990), 359–370.
  • C. J. Earle, “The genus two Jacobians that are isomorphic to a product of elliptic curves”, pp. 27–36 in The geometry of Riemann surfaces and abelian varieties, edited by J. M. Muñoz Porras et al., Contemp. Math. 397, Amer. Math. Soc., Providence, RI, 2006.
  • E. W. Howe, F. Leprévost, and B. Poonen, “Large torsion subgroups of split Jacobians of curves of genus two or three”, Forum Math. 12:3 (2000), 315–364.
  • G. J. Janusz, “Simple components of $Q[{\rm SL}(2,\,q)]$”, Comm. Algebra 1 (1974), 1–22.
  • E. Kani and M. Rosen, “Idempotent relations and factors of Jacobians”, Math. Ann. 284:2 (1989), 307–327.
  • G. Karpilovsky, Group representations, vol. 3, North-Holland Mathematics Studies 180, North-Holland Publishing Co., Amsterdam, 1994.
  • M. Kuwata, “Quadratic twists of an elliptic curve and maps from a hyperelliptic curve”, Math. J. Okayama Univ. 47 (2005), 85–97.
  • A. M. Macbeath, “Generators of the linear fractional groups”, pp. 14–32 in Number Theory (Houston, Tex., 1967), vol. XII, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., 1969.
  • K. Magaard, T. Shaska, and H. V ölklein, “Genus 2 curves that admit a degree 5 map to an elliptic curve”, Forum Math. 21:3 (2009), 547–566.
  • J. S. Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, 1980.
  • J. Paulhus, “Decomposing Jacobians of curves with extra automorphisms”, Acta Arith. 132:3 (2008), 231–244.
  • K. Rubin and A. Silverberg, “Rank frequencies for quadratic twists of elliptic curves”, Experiment. Math. 10:4 (2001), 559–569.
  • J. Wolfart, “Regular dessins, endomorphisms of Jacobians, and transcendence”, pp. 107–120 in A panorama of number theory or the view from Baker's garden (Zürich, 1999), edited by G. Wüstholz, Cambridge Univ. Press, 2002.