# American Institute of Mathematical Sciences

January  2014, 21: 132-136. doi: 10.3934/era.2014.21.132

## An arithmetic ball quotient surface whose Albanese variety is not of CM type

 1 Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, United States

Received  October 2013 Revised  May 2014 Published  September 2014

An example is given of a compact quotient of the unit ball in $\mathbb{C}^2$ by an arithmetic group acting freely such that the Albanese variety is not of CM type. Such examples do not exist for congruence subgroups.
Citation: Chad Schoen. An arithmetic ball quotient surface whose Albanese variety is not of CM type. Electronic Research Announcements, 2014, 21: 132-136. doi: 10.3934/era.2014.21.132
##### References:
 [1] T. Chinburg and M. Stover, Arizona winter school course lecture notes,, 2012. Available from: \url{http://swc.math.arizona.edu/aws/2012/index.html}., (). [2] D. Cox, Primes of the Form $x^2+ny^2$,, A Wiley-Interscience Publication, (1989). [3] J. Cremona, Algorithms for Modular Elliptic Curves,, Second edition, (1997). [4] F. Diamond and J. Shurman, A First Course in Modular Forms,, Graduate Texts in Mathematics, (2005). [5] N. Elkies, The Klein Quartic in Number Theory,, in \emph{The Eightfold Way}, (1999), 51. [6] R. Hartshorne, Algebraic Geometry,, Graduate Texts in Mathematics, (1977). [7] F. Hirzebruch, Arrangements of lines and algebraic surfaces,, \emph{Arithmetic and Geometry, (1983), 113. [8] M. Inoue, Some new surfaces of general type,, \emph{Tokyo J. Math.}, 17 (1994), 295. doi: 10.3836/tjm/1270127954. [9] M.-N. Ishida, The irregularities of Hirzebruch's examples of surfaces of general type with $c_1^2=3c_2$,, \emph{Math. Ann.}, 262 (1983), 407. doi: 10.1007/BF01456018. [10] S. Lang, Abelain Varieties,, Interscience Tracts in Pure and Applied Mathematics. No. 7, (1959). [11] R. Livné, On Certain Covers of the Universal Elliptic Curve,, Ph.D. Thesis, (1981). [12] Y. Miyoaka, The maximal number of quotients singularities on surfaces with given numerical invariants,, \emph{Math. Ann.}, 268 (1984), 159. doi: 10.1007/BF01456083. [13] K. Murty and D. Ramakrishnan, The Albanese of unitary Shimura varieties,, in \emph{The Zeta Function of Picard Modular Surfaces} (eds. R. Langlands and D. Ramakrishnan), (1992), 445. [14] J. D. Rogawski, Analytic expression for the number of points mod $p$,, in \emph{The Zeta Function of Picard Modular Surfaces} (eds. R. Langlands and D. Ramakrishnan), (1992), 65. [15] J. Silverman, The Arithmetic of Elliptic Curves,, Graduate Texts in Mathematics, (1986). doi: 10.1007/978-1-4757-1920-8. [16] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves,, Graduate Texts in Mathematics, (1994). doi: 10.1007/978-1-4612-0851-8. [17] R. O. Wells, Differential Analysis on Complex Manifolds,, Prentice-Hall Series in Modern Analysis, (1973). [18] T. Yamazaki and M. Yoshida, On Hirzebruch's examples of surfaces with $c_1^2=3c_2$,, \emph{Math. Ann.}, 266 (1984), 421. doi: 10.1007/BF01458537. [19] S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry,, \emph{Proc. Natl. Acad. Sci. USA}, 74 (1977), 1798. doi: 10.1073/pnas.74.5.1798.

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##### References:
 [1] T. Chinburg and M. Stover, Arizona winter school course lecture notes,, 2012. Available from: \url{http://swc.math.arizona.edu/aws/2012/index.html}., (). [2] D. Cox, Primes of the Form $x^2+ny^2$,, A Wiley-Interscience Publication, (1989). [3] J. Cremona, Algorithms for Modular Elliptic Curves,, Second edition, (1997). [4] F. Diamond and J. Shurman, A First Course in Modular Forms,, Graduate Texts in Mathematics, (2005). [5] N. Elkies, The Klein Quartic in Number Theory,, in \emph{The Eightfold Way}, (1999), 51. [6] R. Hartshorne, Algebraic Geometry,, Graduate Texts in Mathematics, (1977). [7] F. Hirzebruch, Arrangements of lines and algebraic surfaces,, \emph{Arithmetic and Geometry, (1983), 113. [8] M. Inoue, Some new surfaces of general type,, \emph{Tokyo J. Math.}, 17 (1994), 295. doi: 10.3836/tjm/1270127954. [9] M.-N. Ishida, The irregularities of Hirzebruch's examples of surfaces of general type with $c_1^2=3c_2$,, \emph{Math. Ann.}, 262 (1983), 407. doi: 10.1007/BF01456018. [10] S. Lang, Abelain Varieties,, Interscience Tracts in Pure and Applied Mathematics. No. 7, (1959). [11] R. Livné, On Certain Covers of the Universal Elliptic Curve,, Ph.D. Thesis, (1981). [12] Y. Miyoaka, The maximal number of quotients singularities on surfaces with given numerical invariants,, \emph{Math. Ann.}, 268 (1984), 159. doi: 10.1007/BF01456083. [13] K. Murty and D. Ramakrishnan, The Albanese of unitary Shimura varieties,, in \emph{The Zeta Function of Picard Modular Surfaces} (eds. R. Langlands and D. Ramakrishnan), (1992), 445. [14] J. D. Rogawski, Analytic expression for the number of points mod $p$,, in \emph{The Zeta Function of Picard Modular Surfaces} (eds. R. Langlands and D. Ramakrishnan), (1992), 65. [15] J. Silverman, The Arithmetic of Elliptic Curves,, Graduate Texts in Mathematics, (1986). doi: 10.1007/978-1-4757-1920-8. [16] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves,, Graduate Texts in Mathematics, (1994). doi: 10.1007/978-1-4612-0851-8. [17] R. O. Wells, Differential Analysis on Complex Manifolds,, Prentice-Hall Series in Modern Analysis, (1973). [18] T. Yamazaki and M. Yoshida, On Hirzebruch's examples of surfaces with $c_1^2=3c_2$,, \emph{Math. Ann.}, 266 (1984), 421. doi: 10.1007/BF01458537. [19] S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry,, \emph{Proc. Natl. Acad. Sci. USA}, 74 (1977), 1798. doi: 10.1073/pnas.74.5.1798.
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