The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields

Abstract

We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.

  • [Abramovich-Voloch1992] Go to document D. Abramovich and J. F. Voloch, "Toward a proof of the Mordell-Lang conjecture in characteristic $p$," Internat. Math. Res. Notices, vol. 5, iss. 5, pp. 103-115, 1992.
    @article {Abramovich-Voloch1992, MRKEY = {1162230},
      AUTHOR = {Abramovich, Dan and Voloch, Jos{é} Felipe},
      TITLE = {Toward a proof of the {M}ordell-{L}ang conjecture in characteristic {$p$}},
      JOURNAL = {Internat. Math. Res. Notices},
      VOLUME={5},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {1992},
      NUMBER = {5},
      PAGES = {103--115},
      ISSN = {1073-7928},
      MRCLASS = {11G10 (14K05)},
      MRNUMBER = {94f:11051},
      DOI = {10.1155/S1073792892000126},
      ZBLNUMBER = {0787.14026},
      }
  • [BourbakiAlgebra4-7] N. Bourbaki, Algebra II. Chapters 4–7, New York: Springer-Verlag, 2003.
    @book {BourbakiAlgebra4-7, MRKEY = {1994218},
      AUTHOR = {Bourbaki, Nicolas},
      TITLE = {Algebra {\rm II}. {C}hapters 4--7},
      SERIES = {Elements of Mathematics (Berlin)},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2003},
      PAGES = {viii+461},
      ISBN = {3-540-00706-7},
      MRCLASS = {00A05 (12-01 13-01)},
      MRNUMBER = {1994218},
      ZBLNUMBER = {1017.12001},
      }
  • [BourbakiCommutativeAlgebra] N. Bourbaki, Commutative Algebra, New York: Springer-Verlag, 1998.
    @book{BourbakiCommutativeAlgebra, KEY={Bou98},
      AUTHOR={Bourbaki, Nicolas},
      TITLE={Commutative Algebra},
      PUBLISHER={Springer-Verlag},
      ADDRESS={New York},
      YEAR={1998},
      }
  • [Buium-Voloch1993] Go to document A. Buium and J. F. Voloch, "Integral points of abelian varieties over function fields of characteristic zero," Math. Ann., vol. 297, iss. 2, pp. 303-307, 1993.
    @article {Buium-Voloch1993, MRKEY = {1241808},
      AUTHOR = {Buium, Alexandru and Voloch, Jos{é} Felipe},
      TITLE = {Integral points of abelian varieties over function fields of characteristic zero},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {297},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {303--307},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14G05 (14K05 14L05)},
      MRNUMBER = {94i:14029},
      MRREVIEWER = {Joseph H. Silverman},
      DOI = {10.1007/BF01459503},
      ZBLNUMBER = {0789.14017},
      }
  • [Cassels1962] Go to document J. W. S. Cassels, "Arithmetic on curves of genus $1$. III. The Tate-\v Safarevič and Selmer groups," Proc. London Math. Soc., vol. 12, pp. 259-296, 1962.
    @article {Cassels1962, MRKEY = {0163913},
      AUTHOR = {Cassels, J. W. S.},
      TITLE = {Arithmetic on curves of genus {$1$}. {III}. {T}he {T}ate-\v {S}afarevič and {S}elmer groups},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {12},
      YEAR = {1962},
      PAGES = {259--296},
      ISSN = {0024-6115},
      MRCLASS = {14.49 (14.51)},
      MRNUMBER = {29 \#1212},
      MRREVIEWER = {J. Gu{é}rindon},
      DOI = {10.1112/plms/s3-12.1.259},
      }
  • [Cassels1964] J. W. S. Cassels, "Arithmetic on curves of genus $1$. VII. The dual exact sequence," J. Reine Angew. Math., vol. 216, pp. 150-158, 1964.
    @article {Cassels1964, MRKEY = {0169849},
      AUTHOR = {Cassels, J. W. S.},
      TITLE = {Arithmetic on curves of genus {$1$}. {VII}. {T}he dual exact sequence},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {216},
      YEAR = {1964},
      PAGES = {150--158},
      ISSN = {0075-4102},
      MRCLASS = {14.40 (10.66)},
      MRNUMBER = {30 \#92},
      MRREVIEWER = {T. Ono},
      ZBLNUMBER = {0146.42304},
      }
  • [Delon1998] F. Delon, "Separably closed fields," in Model Theory and Algebraic Geometry, New York: Springer-Verlag, 1998, pp. 143-176.
    @incollection {Delon1998, MRKEY = {1678543},
      AUTHOR = {Delon, Fran{ç}oise},
      TITLE = {Separably closed fields},
      BOOKTITLE = {Model Theory and Algebraic Geometry},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {1696},
      PAGES = {143--176},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1998},
      MRCLASS = {12L12 (03C45 03C60)},
      MRNUMBER = {2000a:12011},
      MRREVIEWER = {Ricardo Bianconi},
      ZBLNUMBER = {0925.03169},
      }
  • [GonzalezAviles-Tan-preprint2006] C. D. González-Avilés and K. Tan, "A generalization of the Cassels-Tate dual exact sequence," Math. Res. Lett., vol. 14, pp. 295-302, 2007.
    @article{GonzalezAviles-Tan-preprint2006, KEY={GAT07},
      AUTHOR={González-Avilés, Cristian D. and Tan, Ki-Seng},
      TITLE={A generalization of the Cassels-Tate dual exact sequence},
      JOURNAL={Math. Res. Lett.},
      VOLUME={14},
      YEAR={2007},
      PAGES={295--302},
      }
  • [Flynn2004] Go to document E. V. Flynn, "The Hasse principle and the Brauer-Manin obstruction for curves," Manuscripta Math., vol. 115, iss. 4, pp. 437-466, 2004.
    @article {Flynn2004, MRKEY = {2103661},
      AUTHOR = {Flynn, E. V.},
      TITLE = {The {H}asse principle and the {B}rauer-{M}anin obstruction for curves},
      JOURNAL = {Manuscripta Math.},
      FJOURNAL = {Manuscripta Mathematica},
      VOLUME = {115},
      YEAR = {2004},
      NUMBER = {4},
      PAGES = {437--466},
      ISSN = {0025-2611},
      CODEN = {MSMHB2},
      MRCLASS = {11G30 (14G05 14H25 14H40)},
      MRNUMBER = {2005j:11047},
      MRREVIEWER = {Tam{á}s Szamuely},
      DOI = {10.1007/s00229-004-0502-9},
      ZBLNUMBER = {1069.11023},
      }
  • [Greenberg1966] Go to document M. J. Greenberg, "Rational points in Henselian discrete valuation rings," Inst. Hautes Études Sci. Publ. Math., vol. 31, pp. 59-64, 1966.
    @article {Greenberg1966, MRKEY = {0207700},
      AUTHOR = {Greenberg, Marvin J.},
      TITLE = {Rational points in {H}enselian discrete valuation rings},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {31},
      YEAR = {1966},
      PAGES = {59--64},
      ISSN = {0073-8301},
      MRCLASS = {13.98 (14.40)},
      MRNUMBER = {34 \#7515},
      MRREVIEWER = {H. Yanagihara},
      URL = {http://www.numdam.org/item?id=PMIHES_1966__31__59_0},
      ZBLNUMBER = {0146.42201},
      }
  • [Grothendieck-Brauer3] A. Grothendieck, "Le groupe de Brauer. III. Exemples et compléments," in Dix Exposés sur la Cohomologie des Schémas, Amsterdam: North-Holland, 1968, pp. 88-188.
    @incollection {Grothendieck-Brauer3, MRKEY = {0244271},
      AUTHOR = {Grothendieck, Alexander},
      TITLE = {Le groupe de {B}rauer. {III}. {E}xemples et compléments},
      BOOKTITLE = {Dix {E}xposés sur la {C}ohomologie des {S}chémas},
      PAGES = {88--188},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {1968},
      MRCLASS = {14.55},
      MRNUMBER = {39 \#5586c},
      MRREVIEWER = {J. S. Milne},
      ZBLNUMBER = {0198.25901},
      }
  • [Hrushovski1996] Go to document E. Hrushovski, "The Mordell-Lang conjecture for function fields," J. Amer. Math. Soc., vol. 9, iss. 3, pp. 667-690, 1996.
    @article {Hrushovski1996, MRKEY = {1333294},
      AUTHOR = {Hrushovski, Ehud},
      TITLE = {The {M}ordell-{L}ang conjecture for function fields},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {9},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {667--690},
      ISSN = {0894-0347},
      MRCLASS = {11U09 (03C60 11G10 14G99)},
      MRNUMBER = {97h:11154},
      MRREVIEWER = {Anand Pillay},
      DOI = {10.1090/S0894-0347-96-00202-0},
      ZBLNUMBER = {0864.03026},
      }
  • [LangFundamentals] S. Lang, Fundamentals of Diophantine Geometry, New York: Springer-Verlag, 1983, vol. 191.
    @book {LangFundamentals, MRKEY = {715605},
      AUTHOR = {Lang, Serge},
      TITLE = {Fundamentals of {D}iophantine Geometry},
      SERIES={Grad. Texts. Math.},
      VOLUME={191},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1983},
      PAGES = {xviii+370},
      ISBN = {0-387-90837-4},
      MRCLASS = {11-02 (11Dxx 11Gxx 14G25)},
      MRNUMBER = {85j:11005},
      MRREVIEWER = {Gerd Faltings},
      ZBLNUMBER = {0528.14013},
      }
  • [Lang-Neron1959] Go to document S. Lang and A. Néron, "Rational points of abelian varieties over function fields," Amer. J. Math., vol. 81, pp. 95-118, 1959.
    @article {Lang-Neron1959, MRKEY = {0102520},
      AUTHOR = {Lang, S. and N{é}ron, A.},
      TITLE = {Rational points of abelian varieties over function fields},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {81},
      YEAR = {1959},
      PAGES = {95--118},
      ISSN = {0002-9327},
      MRCLASS = {14.00},
      MRNUMBER = {21 \#1311},
      MRREVIEWER = {M. Nagata},
      DOI = {10.2307/2372851},
      ZBLNUMBER = {0099.16103},
      }
  • [Manin1963] J. I. Manin, "Rational points on algebraic curves over function fields," Izv. Akad. Nauk SSSR Ser. Mat., vol. 27, pp. 1395-1440, 1963.
    @article {Manin1963, MRKEY = {0157971},
      AUTHOR = {Manin, Ju. I.},
      TITLE = {Rational points on algebraic curves over function fields},
      JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {27},
      YEAR = {1963},
      PAGES = {1395--1440},
      ISSN = {0373-2436},
      MRCLASS = {14.40 (10.10)},
      MRNUMBER = {28 \#1199},
      MRREVIEWER = {S. Lang},
      ZBLNUMBER = {0178.55102},
      }
  • [Manin1971] Y. I. Manin, "Le groupe de Brauer-Grothendieck en géométrie diophantienne," in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Paris: Gauthier-Villars, 1971, pp. 401-411.
    @incollection {Manin1971, MRKEY = {0427322},
      AUTHOR = {Manin, Y. I.},
      TITLE = {Le groupe de {B}rauer-{G}rothendieck en géométrie diophantienne},
      BOOKTITLE = {Actes du {C}ongrès {I}nternational des {M}athématiciens ({N}ice, 1970), {T}ome 1},
      PAGES = {401--411},
      PUBLISHER = {Gauthier-Villars},
      ADDRESS = {Paris},
      YEAR = {1971},
      MRCLASS = {14G25},
      MRNUMBER = {55 \#356},
      MRREVIEWER = {Loren D. Olson},
      ZBLNUMBER = {0239.14010},
      }
  • [Manin1989] J. I. Manin, "Rational points on algebraic curves over function fields," Izv. Akad. Nauk SSSR Ser. Mat., vol. 27, pp. 1395-1440, 1963.
    @article {Manin1989, MRKEY = {0157971},
      AUTHOR = {Manin, Ju. I.},
      TITLE = {Rational points on algebraic curves over function fields},
      JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {27},
      YEAR = {1963},
      PAGES = {1395--1440},
      ISSN = {0373-2436},
      MRCLASS = {14.40 (10.10)},
      MRNUMBER = {28 \#1199},
      MRREVIEWER = {S. Lang},
      ZBLNUMBER = {0178.55102},
      }
  • [Milne1970b] Go to document J. S. Milne, "Elements of order $p$ in the Tate-\v Safarevič group," Bull. London Math. Soc., vol. 2, pp. 293-296, 1970.
    @article {Milne1970b, MRKEY = {0277507},
      AUTHOR = {Milne, J. S.},
      TITLE = {Elements of order {$p$} in the {T}ate-\v {S}afarevič group},
      JOURNAL = {Bull. London Math. Soc.},
      FJOURNAL = {The Bulletin of the London Mathematical Society},
      VOLUME = {2},
      YEAR = {1970},
      PAGES = {293--296},
      ISSN = {0024-6093},
      MRCLASS = {12.50 (10.00)},
      MRNUMBER = {43 \#3240},
      MRREVIEWER = {F. Oort},
      DOI = {10.1112/blms/2.3.293},
      }
  • [Milne1972b] J. S. Milne, "Congruence subgroups of abelian varieties," Bull. Sci. Math., vol. 96, pp. 333-338, 1972.
    @article {Milne1972b, MRKEY = {0321936},
      AUTHOR = {Milne, J. S.},
      TITLE = {Congruence subgroups of abelian varieties},
      JOURNAL = {Bull. Sci. Math.},
      FJOURNAL = {Bulletin des Sciences Mathématiques. 2e Série},
      VOLUME = {96},
      YEAR = {1972},
      PAGES = {333--338},
      ISSN = {0007-4497},
      MRCLASS = {14K15},
      MRNUMBER = {48 \#301},
      MRREVIEWER = {Michiel Hazewinkel},
      ZBLNUMBER = {0247.14003},
      }
  • [MilneEtale] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980.
    @book {MilneEtale, MRKEY = {559531},
      AUTHOR = {Milne, J. S.},
      TITLE = {\'{E}tale Cohomology},
      SERIES = {Princeton Math. Series},
      NUMBER = {33},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1980},
      PAGES = {xiii+323},
      ISBN = {0-691-08238-3},
      MRCLASS = {14-02 (14F20 18F99)},
      MRNUMBER = {81j:14002},
      MRREVIEWER = {G. Horrocks},
      ZBLNUMBER = {0433.14012},
      }
  • [MilneADT] J. S. Milne, Arithmetic Duality Theorems, Boston, MA: Academic Press, Inc., 1986.
    @book {MilneADT, MRKEY = {881804},
      AUTHOR = {Milne, J. S.},
      TITLE = {Arithmetic Duality Theorems},
      SERIES = {Perspect. Math.},
      NUMBER = {1},
      PUBLISHER = {Academic Press, Inc.},
      ADDRESS = {Boston, MA},
      YEAR = {1986},
      PAGES = {x+421},
      ISBN = {0-12-498040-6},
      MRCLASS = {14F20 (11G99 11R34 12G10)},
      MRNUMBER = {88e:14028},
      MRREVIEWER = {Gerd Faltings},
      ZBLNUMBER = {0613.14019},
      }
  • [Poonen-heuristic2006] Go to document B. Poonen, "Heuristics for the Brauer-Manin obstruction for curves," Experiment. Math., vol. 15, iss. 4, pp. 415-420, 2006.
    @article {Poonen-heuristic2006, MRKEY = {2293593},
      AUTHOR = {Poonen, Bjorn},
      TITLE = {Heuristics for the {B}rauer-{M}anin obstruction for curves},
      JOURNAL = {Experiment. Math.},
      FJOURNAL = {Experimental Mathematics},
      VOLUME = {15},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {415--420},
      ISSN = {1058-6458},
      MRCLASS = {11G30 (11G10 14G05)},
      MRNUMBER = {2008d:11062},
      MRREVIEWER = {Matthew H. Baker},
      URL = {http://projecteuclid.org/getRecord?id=euclid.em/1175789777},
      ZBLNUMBER = {05196200},
      }
  • [Scharaschkin-thesis] V. Scharaschkin, "Local-global problems and the Brauer-Manin obstruction," PhD Thesis , University of Michigan, 1999.
    @phdthesis{Scharaschkin-thesis, KEY={Sch99},
      AUTHOR={ Scharaschkin, Victor},
      TITLE={Local-global problems and the Brauer-Manin obstruction},
      SCHOOL={University of Michigan},
      YEAR={1999},
      }
  • [Serre1964] J. Serre, "Sur les groupes de congruence des variétés abéliennes," Izv. Akad. Nauk SSSR Ser. Mat., vol. 28, pp. 3-20, 1964.
    @article {Serre1964, MRKEY = {0160783},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Sur les groupes de congruence des variétés abéliennes},
      JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {28},
      YEAR = {1964},
      PAGES = {3--20},
      ISSN = {0373-2436},
      MRCLASS = {14.49 (14.51)},
      MRNUMBER = {28 \#3994},
      MRREVIEWER = {J. W. S. Cassels},
      ZBLNUMBER = {0128.15601},
      }
  • [Serre1971] J. Serre, "Sur les groupes de congruence des variétés abéliennes. II," Izv. Akad. Nauk SSSR Ser. Mat., vol. 35, pp. 731-737, 1971.
    @article {Serre1971, MRKEY = {0289513},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Sur les groupes de congruence des variétés abéliennes. {II}},
      JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {35},
      YEAR = {1971},
      PAGES = {731--737},
      ISSN = {0373-2436},
      MRCLASS = {14.05 (10.00)},
      MRNUMBER = {44 \#6702},
      MRREVIEWER = {J. W. S. Cassels},
      ZBLNUMBER = {0222.14025},
      }
  • [SerreLocalFields] J. Serre, Local Fields, New York: Springer-Verlag, 1979.
    @book {SerreLocalFields, MRKEY = {554237},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Local Fields},
      SERIES = {Grad. Texts Math.},
      NUMBER = {67},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1979},
      PAGES = {viii+241},
      ISBN = {0-387-90424-7},
      MRCLASS = {12Bxx},
      MRNUMBER = {82e:12016},
      ZBLNUMBER = {0423.12016},
      }
  • [SerreLieAlgebrasLieGroups] J. Serre, Lie Algebras and Lie Groups, Second ed., New York: Springer-Verlag, 1992.
    @book {SerreLieAlgebrasLieGroups, MRKEY = {1176100},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Lie Algebras and {L}ie Groups},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {1500},
      EDITION = {Second},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1992},
      PAGES = {viii+168},
      ISBN = {3-540-55008-9},
      MRCLASS = {17-01 (17Bxx 22-01)},
      MRNUMBER = {93h:17001},
      ZBLNUMBER = {0742.17008},
      }
  • [Skorobogatov2001] A. Skorobogatov, Torsors and Rational Points, Cambridge: Cambridge Univ. Press, 2001.
    @book {Skorobogatov2001, MRKEY = {1845760},
      AUTHOR = {Skorobogatov, Alexei},
      TITLE = {Torsors and Rational Points},
      SERIES = {Cambridge Tracts in Math.},
      NUMBER = {144},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2001},
      PAGES = {viii+187},
      ISBN = {0-521-80237-7},
      MRCLASS = {14G05 (11G35 11S25 14D10 14G25 14L30)},
      MRNUMBER = {2002d:14032},
      MRREVIEWER = {Tam{á}s Szamuely},
      ZBLNUMBER = {0972.14015},
      }
  • [Stoll2006preprint] M. Stoll, "Finite descent and rational points on curves," Algebra and Number Theory, vol. 1, pp. 349-391, 2007.
    @article{Stoll2006preprint, KEY={Sto07},
      AUTHOR={Stoll, Michael},
      TITLE={Finite descent and rational points on curves},
      JOURNAL={Algebra and Number Theory},
      VOLUME={1},
      YEAR={2007},
      PAGES={349--391},
      ISSN={1937-0652},
      MRCLASS={11G30 (11G10 14G05)},
      MRNUMBER={2008i:11086},
      MRREVIEWER={Pavlos Tzermias},
      }
  • [Voloch1995] Go to document J. F. Voloch, "Diophantine approximation on abelian varieties in characteristic $p$," Amer. J. Math., vol. 117, iss. 4, pp. 1089-1095, 1995.
    @article {Voloch1995, MRKEY = {1342843},
      AUTHOR = {Voloch, Jos{é} Felipe},
      TITLE = {Diophantine approximation on abelian varieties in characteristic {$p$}},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {117},
      YEAR = {1995},
      NUMBER = {4},
      PAGES = {1089--1095},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {11G10 (11J99 14K15)},
      MRNUMBER = {96i:11061},
      MRREVIEWER = {Alexandru Buium},
      DOI = {10.2307/2374961},
      ZBLNUMBER = {0855.11029},
      }
  • [Wang1996] Go to document L. Wang, "Brauer-Manin obstruction to weak approximation on abelian varieties," Israel J. Math., vol. 94, pp. 189-200, 1996.
    @article {Wang1996, MRKEY = {1394574},
      AUTHOR = {Wang, Lan},
      TITLE = {Brauer-{M}anin obstruction to weak approximation on abelian varieties},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {94},
      YEAR = {1996},
      PAGES = {189--200},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {11G10 (14K15)},
      MRNUMBER = {97e:11069},
      MRREVIEWER = {Enric Nart},
      DOI = {10.1007/BF02762704},
      ZBLNUMBER = {0870.14032},
      }
  • [Zarhin2006preprint] Y. G. Zarhin, "Abelian varieties without homotheties," Math. Res. Lett., vol. 14, iss. 1, pp. 157-164, 2007.
    @article {Zarhin2006preprint, MRKEY = {2289628},
      AUTHOR = {Zarhin, Yuri G.},
      TITLE = {Abelian varieties without homotheties},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {14},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {157--164},
      ISSN = {1073-2780},
      MRCLASS = {11G10 (14K15 14K22)},
      MRNUMBER = {2007j:11077},
      MRREVIEWER = {Ivica Gusi{ć}},
      ZBLNUMBER = {1128.11031},
      }

Authors

Bjorn Poonen

Massachusetts Institute of Technology
Department of Mathematics
Building 2, Room 244
77 Massachusetts Avenue
Cambridge MA 02139-4307
United States

José Felipe Voloch

The University of Texas at Austin
Department of Mathematics
1 University Station C1200
Austin TX 78712
United States