Abstract
We construct absolutely simple jacobians of nonhyperelliptic genus 4 curves, using Del Pezzo surfaces of degree 1.
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Bourbaki N. (1968). Groupes et algèbres de Lie, Chapitres IV, V, VI. Hermann, Paris
Conway J.H., Curtis R.T., Norton S.P., Parker R.A. and Wilson R.A. (1985). Atlas of Finite Groups. Clarendon Press, Oxford
Demazure, M.: Surfaces de Del Pezzo II, III, IV, V. In: Demazure, M., Pinkham, H., Teissier, B., (eds.) Séminaire sur les singularités des surfaces. Springer Lecture Notes in Math., vol. 777, pp. 23–69 (1980)
Dolgachev, I., Ortland, D.: Point sets in projective spaces and theta functions. Astérisque, vol. 165 (1986)
Dolgachev, I.: Topics in classical agebraic geometry, Part 1, Chapters 1–11. Available at http://www. math.lsa.umich.edu/~idolga/lecturenotes.html
Ekedahl, T.: An effective version of Hilbert’s irreducibility theorem. In: Séminaire de Théorie des Nombres, Paris 1988–89, pp. 241–249. Birkhäuser, Boston (1990)
Elkin A. and Zarhin Y.G. (2006). Endomorphism algebras of hyperelliptic Jacobians and finite projective lines. J. Ramanujan Math. Soc. 21: 169–187, arXiv:math.AG/0508120
Erné R. (1994). Construction of a del Pezzo surface with maximal Galois action on its Picard group. J. Pure Appl. Algebra 97: 15–27
Feit W. and Tits J. (1978). Projective representations of minimum degree of group extensions. Canad. J. Math. 30: 1092–1102
Hartshorne R. (1977). Algebraic Geometry, GTM, vol. 52. Springer, Heidelberg
Iskovskikh V.A. (1979). Minimal models of rational surfaces over arbitrary fields. Izv. Akad. Nauk Ser. Mat. 43: 19–43
Iskovskikh V.A. (1980). Minimal models of rational surfaces over arbitrary fields. Math. USSR-Izv. 14: 17–39
Jansen Ch., Lux Kl., Parker R. and Wilson R. (1995). The Atlas of Brauer Characters. Oxford University Press, Oxford
Kleidman P.B. and Liebeck M.W. (1989). On a theorem of Feit and Tits. Proc. Am. Math. Soc. 107: 315–322
Lang S. Introduction to Algebraic and Abelian Functions, 2nd edn. GTM, vol. 89. Springer, New York (1982)
Manin Y.I. (1967). Rational surfaces over perfect fields. II. Mat. Sb. (N.S.) 72(114): 161–192
Manin Y.I. (1967). Rational surfaces over perfect fields. Math. USSR Sbornik 1: 141–168
Manin Y.I. (1986). Cubic Forms, 2nd edn. North Holland, Amsterdam
Mortimer B. (1980). The modular permutation representations of the known doubly transitive groups. Proc. Lond. Math. Soc. 41(3): 1–20
Mumford D. (1974). Abelian Varieties, 2nd edn. Oxford University Press, Oxford
Mumford D. (1971). Theta characteristics of an algebraic curve. Ann. Sci. ENS 4(4): 181–192
Oort F. (1988). Endomorphism algebras of abelian varieties. In: Hijikata, H. (eds) Algebraic Geometry and Commutative Algebra in Honor of M. Nagata, vol. II, pp 469–502. Kinokuniya Cy, Tokyo
Passman D.S. (1968). Permutation groups. W.A. Benjamin, Inc., New York
Silverberg A. (1992). Fields of definition for homomorphisms of abelian varieties. J. Pure Appl. Algebra 77: 253–262
Serre J.-P. (1989). Lectures on Mordell–Weil Theorem, 2nd edn. F. Viehweg und Sohn, Braunschweig and Wisbaden
Serre J.-P. (1992). Topics in Galois Theory. Jones and Bartlett Publ., Boston and London
Shafarevich I.R. (1994). Basic Algebraic Geometry, vol. 1, 2nd edn. Springer, Heidelberg
Zarhin Y.G. (2000). Hyperelliptic Jacobians without complex multiplication. Math. Res. Lett. 7: 123–132
Zarhin Y.G. (2001). Hyperelliptic Jacobians and modular representations. In: Faber, C. and Oort, F. (eds) Moduli of abelian varieties. Progress in Math., vol. 195, pp 473–490. Birkhäuser, Boston
Zarhin Y.G. (2005). Very simple representations: variations on a theme of Clifford. In: Völklein, H. and Shaska, T. (eds) Progress in Galois Theory, pp 151–168. Springer, Heidelberg
Zarhin, Y.G.: Homomorphisms of abelian varieties. In: Aubry, Y., Lachaud, G., (eds.) Arithmetic, Geometry and Coding Theory (AGCT 2003). Séminaires et Congrés, vol. 11, pp. 189–215 (2005)
Zarhin Y.G. (2005). Del Pezzo surfaces of degree 2 and Jacobians without complex multiplication. Proc. St. Petersburg Math. Soc. 11: 81–91
Zarhin, Y.G. (2005) Del Pezzo surfaces of degree 2 and Jacobians without complex multiplication. AMS Translations—Series 2, vol. 218, pp. 67–75 (2006). arXiv:math.AG/0405156
Zarhin, Y.G.: Endomorphisms of superelliptic Jacobians. arXiv:math.AG/0605028
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Supported by SFB 701 “Spektrale Strukturen und topologische Methoden in der Mathematik” (Fakultät für Mathematik der Universität Bielefeld).
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Zarhin, Y.G. Del Pezzo surfaces of degree 1 and Jacobians. Math. Ann. 340, 407–435 (2008). https://doi.org/10.1007/s00208-007-0157-4
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DOI: https://doi.org/10.1007/s00208-007-0157-4