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

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

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

Chad Schoen ^1,

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

Received October 2013 Revised May 2014 Published September 2014

Abstract
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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.

Keywords: CM type., Albanese variety, Ball quotient surface, complex multiplication.

Mathematics Subject Classification: 11F75 (14J29.

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: http://swc.math.arizona.edu/aws/2012/index.html.
[2]	D. Cox, Primes of the Form $x^2+ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989.
[3]	J. Cremona, Algorithms for Modular Elliptic Curves, Second edition, Cambridge University Press, Cambridge, 1997.
[4]	F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, 228, Springer-Verlag, New York, 2005.
[5]	N. Elkies, The Klein Quartic in Number Theory, in The Eightfold Way, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge, 1999, 51-101.
[6]	R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977.
[7]	F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and Geometry, Vol. II, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983, 113-140.
[8]	M. Inoue, Some new surfaces of general type, Tokyo J. Math., 17 (1994), 295-319. doi: 10.3836/tjm/1270127954.
[9]	M.-N. Ishida, The irregularities of Hirzebruch's examples of surfaces of general type with $c_1^2=3c_2$, Math. Ann., 262 (1983), 407-420. doi: 10.1007/BF01456018.
[10]	S. Lang, Abelain Varieties, Interscience Tracts in Pure and Applied Mathematics. No. 7, Interscience Publishers, Inc., New York, 1959.
[11]	R. Livné, On Certain Covers of the Universal Elliptic Curve, Ph.D. Thesis, Harvard University, 1981.
[12]	Y. Miyoaka, The maximal number of quotients singularities on surfaces with given numerical invariants, Math. Ann., 268 (1984), 159-171. doi: 10.1007/BF01456083.
[13]	K. Murty and D. Ramakrishnan, The Albanese of unitary Shimura varieties, in The Zeta Function of Picard Modular Surfaces (eds. R. Langlands and D. Ramakrishnan), Univ. Montréal, Montréal, 1992, 445-464.
[14]	J. D. Rogawski, Analytic expression for the number of points mod $p$, in The Zeta Function of Picard Modular Surfaces (eds. R. Langlands and D. Ramakrishnan), Univ. Montréal, Montréal, 1992, 65-109.
[15]	J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4757-1920-8.
[16]	J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0851-8.
[17]	R. O. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall Series in Modern Analysis, Prentice Hall, Inc., Englewood Cliffs, NJ, 1973.
[18]	T. Yamazaki and M. Yoshida, On Hirzebruch's examples of surfaces with $c_1^2=3c_2$, Math. Ann., 266 (1984), 421-431. doi: 10.1007/BF01458537.
[19]	S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA, 74 (1977), 1798-1799. 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: http://swc.math.arizona.edu/aws/2012/index.html.
[2]	D. Cox, Primes of the Form $x^2+ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989.
[3]	J. Cremona, Algorithms for Modular Elliptic Curves, Second edition, Cambridge University Press, Cambridge, 1997.
[4]	F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, 228, Springer-Verlag, New York, 2005.
[5]	N. Elkies, The Klein Quartic in Number Theory, in The Eightfold Way, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge, 1999, 51-101.
[6]	R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977.
[7]	F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and Geometry, Vol. II, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983, 113-140.
[8]	M. Inoue, Some new surfaces of general type, Tokyo J. Math., 17 (1994), 295-319. doi: 10.3836/tjm/1270127954.
[9]	M.-N. Ishida, The irregularities of Hirzebruch's examples of surfaces of general type with $c_1^2=3c_2$, Math. Ann., 262 (1983), 407-420. doi: 10.1007/BF01456018.
[10]	S. Lang, Abelain Varieties, Interscience Tracts in Pure and Applied Mathematics. No. 7, Interscience Publishers, Inc., New York, 1959.
[11]	R. Livné, On Certain Covers of the Universal Elliptic Curve, Ph.D. Thesis, Harvard University, 1981.
[12]	Y. Miyoaka, The maximal number of quotients singularities on surfaces with given numerical invariants, Math. Ann., 268 (1984), 159-171. doi: 10.1007/BF01456083.
[13]	K. Murty and D. Ramakrishnan, The Albanese of unitary Shimura varieties, in The Zeta Function of Picard Modular Surfaces (eds. R. Langlands and D. Ramakrishnan), Univ. Montréal, Montréal, 1992, 445-464.
[14]	J. D. Rogawski, Analytic expression for the number of points mod $p$, in The Zeta Function of Picard Modular Surfaces (eds. R. Langlands and D. Ramakrishnan), Univ. Montréal, Montréal, 1992, 65-109.
[15]	J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4757-1920-8.
[16]	J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0851-8.
[17]	R. O. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall Series in Modern Analysis, Prentice Hall, Inc., Englewood Cliffs, NJ, 1973.
[18]	T. Yamazaki and M. Yoshida, On Hirzebruch's examples of surfaces with $c_1^2=3c_2$, Math. Ann., 266 (1984), 421-431. doi: 10.1007/BF01458537.
[19]	S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA, 74 (1977), 1798-1799. doi: 10.1073/pnas.74.5.1798.

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