Joint equidistribution of CM points

Abstract

We prove the mixing conjecture of Michel and Venkatesh for toral packets with negative fundamental discriminants and split at two fixed primes, assuming all splitting fields have no exceptional Landau-Siegel zero. As a consequence we establish for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all of whose components have the same fundamental discriminant, assuming the CM fields are split at two fixed primes and have no exceptional zero.

The joinings theorem of Einsiedler and Lindenstrauss applies to the toral orbits arising in these results. Yet it falls short of demonstrating equidistribution due to the possibility of intermediate algebraic measures supported on Hecke correspondences and their translates.

The main novel contribution is a method to exclude intermediate measures for toral periods. The crux is a geometric expansion of the cross-correlation between the periodic measure on a torus orbit and a Hecke correspondence, expressing it as a short shifted convolution sum. The latter is bounded from above generalizing the method of Shiu and Nair to polynomials in two variables on smooth domains.

  • [Andre] Go to document Y. André, "Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire," J. Reine Angew. Math., vol. 505, pp. 203-208, 1998.
    @ARTICLE{Andre,
      author = {André,
      Yves},
      title = {Finitude des couples d'invariants modulaires singuliers sur une courbe algébrique plane non modulaire},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {505},
      year = {1998},
      pages = {203--208},
      issn = {0075-4102},
      mrclass = {11G15 (11G18)},
      mrnumber = {1662256},
      mrreviewer = {Akio Tamagawa},
      doi = {10.1515/crll.1998.118},
      url = {https://doi.org/10.1515/crll.1998.118},
      zblnumber = {0918.14010},
      }
  • [BurgerSarnak] Go to document M. Burger and P. Sarnak, "Ramanujan duals. II," Invent. Math., vol. 106, iss. 1, pp. 1-11, 1991.
    @ARTICLE{BurgerSarnak,
      author = {Burger, M. and Sarnak, P.},
      title = {Ramanujan duals. {II}},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {106},
      year = {1991},
      number = {1},
      pages = {1--11},
      issn = {0020-9910},
      mrclass = {22E45 (11F70 22E40 22E46)},
      mrnumber = {1123369},
      mrreviewer = {Dipendra Prasad},
      doi = {10.1007/BF01243900},
      url = {https://doi.org/10.1007/BF01243900},
      zblnumber = {0774.11021},
      }
  • [BSR] Go to document J. Bourgain, P. Sarnak, and Z. Rudnick, "Local statistics of lattice points on the sphere," in Modern Trends in Constructive Function Theory, Amer. Math. Soc., Providence, RI, 2016, vol. 661, pp. 269-282.
    @INCOLLECTION{BSR,
      author = {Bourgain, Jean and Sarnak, Peter and Rudnick, Zeév},
      title = {Local statistics of lattice points on the sphere},
      booktitle = {Modern Trends in Constructive Function Theory},
      series = {Contemp. Math.},
      volume = {661},
      pages = {269--282},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2016},
      mrclass = {11K36 (11E12 11E25)},
      mrnumber = {3489563},
      mrreviewer = {Gediminas Stepanauskas},
      doi = {10.1090/conm/661/13287},
      url = {https://doi.org/10.1090/conm/661/13287},
      zblnumber = {1394.11059},
      }
  • [Burgess] Go to document D. A. Burgess, "On character sums and $L$-series," Proc. London Math. Soc. (3), vol. 12, pp. 193-206, 1962.
    @ARTICLE{Burgess,
      author = {Burgess, D. A.},
      title = {On character sums and {$L$}-series},
      journal = {Proc. London Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {12},
      year = {1962},
      pages = {193--206},
      issn = {0024-6115},
      mrclass = {10.41},
      mrnumber = {0132733},
      mrreviewer = {L. Carlitz},
      doi = {10.1112/plms/s3-12.1.193},
      url = {https://doi.org/10.1112/plms/s3-12.1.193},
      zblnumber = {0106.04004},
      }
  • [CasselsQuadraticForms] J. W. S. Cassels, Rational Quadratic Forms, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978, vol. 13.
    @BOOK{CasselsQuadraticForms,
      author = {Cassels, J. W. S.},
      title = {Rational Quadratic Forms},
      series = {London Math. Soc. Monogr.},
      volume = {13},
      publisher = {Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York},
      year = {1978},
      pages = {xvi+413},
      isbn = {0-12-163260-1},
      mrclass = {10C05 (10-01 15A63)},
      mrnumber = {0522835},
      mrreviewer = {Charles J. Parry},
      zblnumber = {0395.10029},
      }
  • [Chelluri] Go to document T. Chelluri, Equidistribution of Roots of Quadratic Congruences, ProQuest LLC, Ann Arbor, MI, 2004.
    @BOOK{Chelluri,
      author = {Chelluri, Thyagaraju},
      title = {Equidistribution of Roots of Quadratic Congruences},
      note = {Thesis (Ph.D.)--Rutgers The State University of New Jersey - New Brunswick},
      publisher = {ProQuest LLC, Ann Arbor, MI},
      year = {2004},
      pages = {33},
      isbn = {978-0496-14210-1},
      mrclass = {Thesis},
      mrnumber = {2717231},
      url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3153552},
      zblnumber = {},
      }
  • [ConreyIwaniec] Go to document J. B. Conrey and H. Iwaniec, "The cubic moment of central values of automorphic $L$-functions," Ann. of Math. (2), vol. 151, iss. 3, pp. 1175-1216, 2000.
    @ARTICLE{ConreyIwaniec,
      author = {Conrey, J. B. and Iwaniec, H.},
      title = {The cubic moment of central values of automorphic {$L$}-functions},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {151},
      year = {2000},
      number = {3},
      pages = {1175--1216},
      issn = {0003-486X},
      mrclass = {11F66 (11F25 11L05 11M06)},
      mrnumber = {1779567},
      mrreviewer = {John B. Friedlander},
      doi = {10.2307/121132},
      url = {https://doi.org/10.2307/121132},
      zblnumber = {0973.11056},
      }
  • [Cornulier] Go to document Y. Cornulier, "On the Chabauty space of locally compact abelian groups," Algebr. Geom. Topol., vol. 11, iss. 4, pp. 2007-2035, 2011.
    @ARTICLE{Cornulier,
      author = {Cornulier, Yves},
      title = {On the {C}habauty space of locally compact abelian groups},
      journal = {Algebr. Geom. Topol.},
      fjournal = {Algebraic \& Geometric Topology},
      volume = {11},
      year = {2011},
      number = {4},
      pages = {2007--2035},
      issn = {1472-2747},
      mrclass = {22B05 (43A25 54D05 54F45)},
      mrnumber = {2826931},
      mrreviewer = {Markus Neuhauser},
      doi = {10.2140/agt.2011.11.2007},
      url = {https://doi.org/10.2140/agt.2011.11.2007},
      zblnumber = {1221.22008},
      }
  • [ClozelOhUllmo] Go to document L. Clozel, H. Oh, and E. Ullmo, "Hecke operators and equidistribution of Hecke points," Invent. Math., vol. 144, iss. 2, pp. 327-351, 2001.
    @ARTICLE{ClozelOhUllmo,
      author = {Clozel, Laurent and Oh, Hee and Ullmo, Emmanuel},
      title = {Hecke operators and equidistribution of {H}ecke points},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {144},
      year = {2001},
      number = {2},
      pages = {327--351},
      issn = {0020-9910},
      mrclass = {11F72},
      mrnumber = {1827734},
      mrreviewer = {Stefan Kühnlein},
      doi = {10.1007/s002220100126},
      url = {https://doi.org/10.1007/s002220100126},
      zblnumber = {1144.11301},
      }
  • [ClozelUllmo] L. Clozel and E. Ullmo, "Équidistribution des points de Hecke," in Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 193-254.
    @INCOLLECTION{ClozelUllmo,
      author = {Clozel, Laurent and Ullmo, Emmanuel},
      title = {\'{E}quidistribution des points de {H}ecke},
      booktitle = {Contributions to Automorphic Forms, Geometry, and Number Theory},
      pages = {193--254},
      publisher = {Johns Hopkins Univ. Press, Baltimore, MD},
      year = {2004},
      mrclass = {11F60 (11F70)},
      mrnumber = {2058609},
      mrreviewer = {A. Raghuram},
      zblnumber = {1068.11042},
      }
  • [DicksonPSLn] Go to document L. E. Dickson, "Theory of linear groups in an arbitrary field," Trans. Amer. Math. Soc., vol. 2, iss. 4, pp. 363-394, 1901.
    @ARTICLE{DicksonPSLn,
      author = {Dickson, Leonard Eugene},
      title = {Theory of linear groups in an arbitrary field},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {2},
      year = {1901},
      number = {4},
      pages = {363--394},
      issn = {0002-9947},
      mrclass = {20G15},
      mrnumber = {1500573},
      doi = {10.2307/1986251},
      url = {https://doi.org/10.2307/1986251},
      zblnumber = {32.0131.03},
      }
  • [DeMilloLipton] Go to document R. A. Demillo and R. J. Lipton, "A probabilistic remark on algebraic program testing," Information Processing Letters, vol. 7, iss. 4, pp. 193-195, 1978.
    @ARTICLE{DeMilloLipton,
      author = {Demillo, Richard A. and J. Lipton, Richard},
      title = {A probabilistic remark on algebraic program testing},
      journal = {Information Processing Letters},
      volume = {7},
      number = {4},
      pages = {193 -- 195},
      year = {1978},
      zblnumber = {0397.68011},
      doi = {10.1016/0020-0190(78)90067-4},
     }
  • [BretecheBrowning] Go to document R. de la Bretèche and T. D. Browning, "Sums of arithmetic functions over values of binary forms," Acta Arith., vol. 125, iss. 3, pp. 291-304, 2006.
    @ARTICLE{BretecheBrowning,
      author = {de la Bretèche, R. and Browning, T. D.},
      title = {Sums of arithmetic functions over values of binary forms},
      journal = {Acta Arith.},
      fjournal = {Acta Arithmetica},
      volume = {125},
      year = {2006},
      number = {3},
      pages = {291--304},
      issn = {0065-1036},
      mrclass = {11N37 (11N32)},
      mrnumber = {2276196},
      mrreviewer = {Ognian Trifonov},
      doi = {10.4064/aa125-3-6},
      url = {https://doi.org/10.4064/aa125-3-6},
      zblnumber = {1159.11035},
      }
  • [BretecheTenenbaum] Go to document R. de la Bretèche and G. Tenenbaum, "Moyennes de fonctions arithmétiques de formes binaires," Mathematika, vol. 58, iss. 2, pp. 290-304, 2012.
    @ARTICLE{BretecheTenenbaum,
      author = {de la Bretèche, Régis and Tenenbaum, Gérald},
      title = {Moyennes de fonctions arithmétiques de formes binaires},
      journal = {Mathematika},
      fjournal = {Mathematika. A Journal of Pure and Applied Mathematics},
      volume = {58},
      year = {2012},
      number = {2},
      pages = {290--304},
      issn = {0025-5793},
      mrclass = {11A25 (11D45 11N37 11N56)},
      mrnumber = {2965973},
      mrreviewer = {Olivier Bordellès},
      doi = {10.1112/S0025579311002154},
      url = {https://doi.org/10.1112/S0025579311002154},
      zblnumber = {1284.11126},
      }
  • [Duke] Go to document W. Duke, "Hyperbolic distribution problems and half-integral weight Maass forms," Invent. Math., vol. 92, iss. 1, pp. 73-90, 1988.
    @ARTICLE{Duke,
      author = {Duke, W.},
      title = {Hyperbolic distribution problems and half-integral weight {M}aass forms},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {92},
      year = {1988},
      number = {1},
      pages = {73--90},
      issn = {0020-9910},
      mrclass = {11F11 (11E32 11F30 11F37)},
      mrnumber = {0931205},
      mrreviewer = {Mark Sheingorn},
      doi = {10.1007/BF01393993},
      url = {https://doi.org/10.1007/BF01393993},
      zblnumber = {0628.10029},
      }
  • [Edixhoven2] Go to document B. Edixhoven, "Special points on the product of two modular curves," Compositio Math., vol. 114, iss. 3, pp. 315-328, 1998.
    @ARTICLE{Edixhoven2,
      author = {Edixhoven, Bas},
      title = {Special points on the product of two modular curves},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {114},
      year = {1998},
      number = {3},
      pages = {315--328},
      issn = {0010-437X},
      mrclass = {11G15 (14G35 14K22)},
      mrnumber = {1665772},
      mrreviewer = {Ben Moonen},
      doi = {10.1023/A:1000539721162},
      url = {https://doi.org/10.1023/A:1000539721162},
      zblnumber = {0928.14019},
      }
  • [EdixhovenArbitrary] Go to document B. Edixhoven, "Special points on products of modular curves," Duke Math. J., vol. 126, iss. 2, pp. 325-348, 2005.
    @ARTICLE{EdixhovenArbitrary,
      author = {Edixhoven, Bas},
      title = {Special points on products of modular curves},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {126},
      year = {2005},
      number = {2},
      pages = {325--348},
      issn = {0012-7094},
      mrclass = {11G18 (11G15 14G35)},
      mrnumber = {2115260},
      mrreviewer = {Mihran Papikian},
      doi = {10.1215/S0012-7094-04-12624-7},
      url = {https://doi.org/10.1215/S0012-7094-04-12624-7},
      zblnumber = {1072.14027},
      }
  • [EKL] Go to document M. Einsiedler, A. Katok, and E. Lindenstrauss, "Invariant measures and the set of exceptions to Littlewood’s conjecture," Ann. of Math. (2), vol. 164, iss. 2, pp. 513-560, 2006.
    @ARTICLE{EKL,
      author = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon},
      title = {Invariant measures and the set of exceptions to {L}ittlewood's conjecture},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {164},
      year = {2006},
      number = {2},
      pages = {513--560},
      issn = {0003-486X},
      mrclass = {22F10 (11J83 28A80 28D15 37A15)},
      mrnumber = {2247967},
      mrreviewer = {Thomas Ward},
      doi = {10.4007/annals.2006.164.513},
      url = {https://doi.org/10.4007/annals.2006.164.513},
      zblnumber = {1109.22004},
      }
  • [Pisa] M. Einsiedler and E. Lindenstrauss, "Diagonal actions on locally homogeneous spaces," in Homogeneous Flows, Moduli Spaces and Arithmetic, Amer. Math. Soc., Providence, RI, 2010, vol. 10, pp. 155-241.
    @INCOLLECTION{Pisa,
      author = {Einsiedler, Manfred and Lindenstrauss, Elon},
      title = {Diagonal actions on locally homogeneous spaces},
      booktitle = {Homogeneous Flows, Moduli Spaces and Arithmetic},
      series = {Clay Math. Proc.},
      volume = {10},
      pages = {155--241},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2010},
      mrclass = {22F30 (22D40 22F10 28D10 28D15 37A15 58J51)},
      mrnumber = {2648695},
      mrreviewer = {Thomas Ward},
      zblnumber = {1225.37005},
      }
  • [EntropySArithmetic] Go to document M. Einsiedler and E. Lindenstrauss, "On measures invariant under tori on quotients of semisimple groups," Ann. of Math. (2), vol. 181, iss. 3, pp. 993-1031, 2015.
    @ARTICLE{EntropySArithmetic,
      author = {Einsiedler, Manfred and Lindenstrauss, Elon},
      title = {On measures invariant under tori on quotients of semisimple groups},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {181},
      year = {2015},
      number = {3},
      pages = {993--1031},
      issn = {0003-486X},
      mrclass = {22E40 (22D40 22F30 37A15 37A17 37A45)},
      mrnumber = {3296819},
      mrreviewer = {Thomas Ward},
      doi = {10.4007/annals.2015.181.3.3},
      url = {https://doi.org/10.4007/annals.2015.181.3.3},
      zblnumber = {1316.22009},
      }
  • [ELJoinings] M. Einsiedler and E. Lindenstrauss, Joinings of higher rank torus actions on homogeneous spaces, 2015.
    @MISC{ELJoinings,
      author = {Einsiedler, Manfred and Lindenstrauss, Elon},
      title = {Joinings of higher rank torus actions on homogeneous spaces},
      year = {2015},
      note = {to appear, \textit{Publ. Math. Inst. Hautes \'{E}tudes Sci.},
      2017},
      arxiv = {1502.05133},
      zblnumber = {},
      }
  • [ELMVPeriodic] Go to document M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, "Distribution of periodic torus orbits on homogeneous spaces," Duke Math. J., vol. 148, iss. 1, pp. 119-174, 2009.
    @ARTICLE{ELMVPeriodic,
      author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
      title = {Distribution of periodic torus orbits on homogeneous spaces},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {148},
      year = {2009},
      number = {1},
      pages = {119--174},
      issn = {0012-7094},
      mrclass = {37A17 (11E57 37A45)},
      mrnumber = {2515103},
      mrreviewer = {Dmitry Y. Kleinbock},
      doi = {10.1215/00127094-2009-023},
      url = {https://doi.org/10.1215/00127094-2009-023},
      zblnumber = {1172.37003},
      }
  • [ELMVCubic] Go to document M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, "Distribution of periodic torus orbits and Duke’s theorem for cubic fields," Ann. of Math. (2), vol. 173, iss. 2, pp. 815-885, 2011.
    @ARTICLE{ELMVCubic,
      author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
      title = {Distribution of periodic torus orbits and {D}uke's theorem for cubic fields},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {173},
      year = {2011},
      number = {2},
      pages = {815--885},
      issn = {0003-486X},
      mrclass = {37A17 (11E57 11M36 22E40 37A45)},
      mrnumber = {2776363},
      mrreviewer = {Valentin Blomer},
      doi = {10.4007/annals.2011.173.2.5},
      url = {https://doi.org/10.4007/annals.2011.173.2.5},
      zblnumber = {1248.37009},
      }
  • [ELMVPGL2] Go to document M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, "The distribution of closed geodesics on the modular surface, and Duke’s theorem," Enseign. Math. (2), vol. 58, iss. 3-4, pp. 249-313, 2012.
    @ARTICLE{ELMVPGL2,
      author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
      title = {The distribution of closed geodesics on the modular surface, and {D}uke's theorem},
      journal = {Enseign. Math. (2)},
      fjournal = {L'Enseignement Mathématique. Revue Internationale. 2e Série},
      volume = {58},
      year = {2012},
      number = {3-4},
      pages = {249--313},
      issn = {0013-8584},
      mrclass = {11F11 (11E16 11F37 14C22 37A45 53C22)},
      mrnumber = {3058601},
      mrreviewer = {Thomas R. Shemanske},
      doi = {10.4171/LEM/58-3-2},
      url = {https://doi.org/10.4171/LEM/58-3-2},
      zblnumber = {1312.37032},
      }
  • [EMV] J. S. Ellenberg, P. Michel, and A. Venkatesh, "Linnik’s ergodic method and the distribution of integer points on spheres," in Automorphic Representations and $L$-Functions, Tata Inst. Fund. Res., Mumbai, 2013, vol. 22, pp. 119-185.
    @INCOLLECTION{EMV,
      author = {Ellenberg, Jordan S. and Michel, Philippe and Venkatesh, Akshay},
      title = {Linnik's ergodic method and the distribution of integer points on spheres},
      booktitle = {Automorphic Representations and {$L$}-Functions},
      series = {Tata Inst. Fundam. Res. Stud. Math.},
      volume = {22},
      pages = {119--185},
      publisher = {Tata Inst. Fund. Res., Mumbai},
      year = {2013},
      mrclass = {11K36 (11E12)},
      mrnumber = {3156852},
      mrreviewer = {Vladimir S. Anashin},
      zblnumber = {1371.11071},
      }
  • [GKM] Go to document A. Granville, D. Koukoulopoulos, and K. Matomäki, "When the sieve works," Duke Math. J., vol. 164, iss. 10, pp. 1935-1969, 2015.
    @ARTICLE{GKM,
      author = {Granville, Andrew and Koukoulopoulos, Dimitris and Matomäki, Kaisa},
      title = {When the sieve works},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {164},
      year = {2015},
      number = {10},
      pages = {1935--1969},
      issn = {0012-7094},
      mrclass = {11N35 (11B30)},
      mrnumber = {3369306},
      mrreviewer = {S\'{a}ndor Z. Kiss},
      doi = {10.1215/00127094-3120891},
      url = {https://doi.org/10.1215/00127094-3120891},
      zblnumber = {1326.11055},
      }
  • [GrossZagier] Go to document B. H. Gross and D. B. Zagier, "Heegner points and derivatives of $L$-series," Invent. Math., vol. 84, iss. 2, pp. 225-320, 1986.
    @ARTICLE{GrossZagier,
      author = {Gross, Benedict H. and Zagier, Don B.},
      title = {Heegner points and derivatives of {$L$}-series},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {84},
      year = {1986},
      number = {2},
      pages = {225--320},
      issn = {0020-9910},
      mrclass = {11G40 (11F11 11G05 14G10)},
      mrnumber = {0833192},
      mrreviewer = {Loren D. Olson},
      doi = {10.1007/BF01388809},
      url = {https://doi.org/10.1007/BF01388809},
      zblnumber = {0608.14019},
      }
  • [HoweMoore] Go to document R. E. Howe and C. C. Moore, "Asymptotic properties of unitary representations," J. Funct. Anal., vol. 32, iss. 1, pp. 72-96, 1979.
    @ARTICLE{HoweMoore,
      author = {Howe, Roger E. and Moore, Calvin C.},
      title = {Asymptotic properties of unitary representations},
      journal = {J. Funct. Anal.},
      fjournal = {Journal of Functional Analysis},
      volume = {32},
      year = {1979},
      number = {1},
      pages = {72--96},
      issn = {0022-1236},
      mrclass = {22E50 (22D10 22E50 43A30 43A80)},
      mrnumber = {0533220},
      mrreviewer = {Jonathan M. Rosenberg},
      doi = {10.1016/0022-1236(79)90078-8},
      url = {https://doi.org/10.1016/0022-1236(79)90078-8},
      zblnumber = {0404.22015},
      }
  • [Holowinsky] R. Holowinsky, "Sieving for mass equidistribution," Ann. of Math. (2), vol. 172, iss. 2, pp. 1499-1516, 2010.
    @ARTICLE{Holowinsky,
      author = {Holowinsky, Roman},
      title = {Sieving for mass equidistribution},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {172},
      year = {2010},
      number = {2},
      pages = {1499--1516},
      issn = {0003-486X},
      mrclass = {11F11 (11F25 11F30 11N36 58J51)},
      mrnumber = {2680498},
      mrreviewer = {Valentin Blomer},
      zblnumber = {1214.11054},
      }
  • [HolowinnskySound] Go to document R. Holowinsky and K. Soundararajan, "Mass equidistribution for Hecke eigenforms," Ann. of Math. (2), vol. 172, iss. 2, pp. 1517-1528, 2010.
    @ARTICLE{HolowinnskySound,
      author = {Holowinsky, Roman and Soundararajan, Kannan},
      title = {Mass equidistribution for {H}ecke eigenforms},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {172},
      year = {2010},
      number = {2},
      pages = {1517--1528},
      issn = {0003-486X},
      mrclass = {11F11 (11F25 11F67 58J51)},
      mrnumber = {2680499},
      mrreviewer = {Valentin Blomer},
      zblnumber = {1211.11050},
      doi = {10.4007/annals.2010/172.1517},
     }
  • [Huxley] Go to document M. N. Huxley, "Exponential sums and lattice points. III," Proc. London Math. Soc. (3), vol. 87, iss. 3, pp. 591-609, 2003.
    @ARTICLE{Huxley,
      author = {Huxley, M. N.},
      title = {Exponential sums and lattice points. {III}},
      journal = {Proc. London Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {87},
      year = {2003},
      number = {3},
      pages = {591--609},
      issn = {0024-6115},
      mrclass = {11L07 (11P21)},
      mrnumber = {2005876},
      mrreviewer = {Angel V. Kumchev},
      doi = {10.1112/S0024611503014485},
      url = {https://doi.org/10.1112/S0024611503014485},
      zblnumber = {1065.11079},
      }
  • [Iwaniec] Go to document H. Iwaniec, "Fourier coefficients of modular forms of half-integral weight," Invent. Math., vol. 87, iss. 2, pp. 385-401, 1987.
    @ARTICLE{Iwaniec,
      author = {Iwaniec, Henryk},
      title = {Fourier coefficients of modular forms of half-integral weight},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {87},
      year = {1987},
      number = {2},
      pages = {385--401},
      issn = {0020-9910},
      mrclass = {11F37 (11F30)},
      mrnumber = {0870736},
      mrreviewer = {Marvin I. Knopp},
      doi = {10.1007/BF01389423},
      url = {https://doi.org/10.1007/BF01389423},
      zblnumber = {0606.10017},
      }
  • [JacquetLangalnds] Go to document H. Jacquet and R. P. Langlands, Automorphic Forms on ${ GL}(2)$, Springer-Verlag, Berlin-New York, 1970, vol. 114.
    @BOOK{JacquetLangalnds,
      author = {Jacquet, H. and Langlands, R. P.},
      title = {Automorphic Forms on {${\rm GL}(2)$}},
      series = {Lecture Notes in Math.},
      volume = {114},
      publisher = {Springer-Verlag, Berlin-New York},
      year = {1970},
      pages = {vii+548},
      mrclass = {10D15 (22E55 12A65 12A70)},
      mrnumber = {0401654},
      mrreviewer = {S. Gelbart},
      zblnumber = {0236.12010},
      doi = {10.1007/BFb0058988},
     }
  • [instability] Go to document G. R. Kempf, "Instability in invariant theory," Ann. of Math. (2), vol. 108, iss. 2, pp. 299-316, 1978.
    @ARTICLE{instability,
      author = {Kempf, George R.},
      title = {Instability in invariant theory},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {108},
      year = {1978},
      number = {2},
      pages = {299--316},
      issn = {0003-486X},
      mrclass = {20G05 (14L30 14M99 15A72)},
      mrnumber = {0506989},
      mrreviewer = {V. L. Popov},
      doi = {10.2307/1971168},
      url = {https://doi.org/10.2307/1971168},
      zblnumber = {0406.14031},
      }
  • [Kh17] Go to document I. Khayutin, "Large deviations and effective equidistribution," Int. Math. Res. Not. IMRN, iss. 10, pp. 3050-3106, 2017.
    @ARTICLE{Kh17,
      author = {Khayutin, Ilya},
      title = {Large deviations and effective equidistribution},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2017},
      number = {10},
      pages = {3050--3106},
      issn = {1073-7928},
      mrclass = {37P55 (37A30 37A50)},
      mrnumber = {3658132},
      mrreviewer = {Thomas Ward},
      doi = {10.1093/imrn/rnw099},
      url = {https://doi.org/10.1093/imrn/rnw099},
      zblnumber = {},
      }
  • [Kneser-p-adic] Go to document M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über ${\germ p}$-adischen Körpern. I," Math. Z., vol. 88, pp. 40-47, 1965.
    @ARTICLE{Kneser-p-adic,
      author = {Kneser, Martin},
      title = {Galois-{K}ohomologie halbeinfacher algebraischer {G}ruppen über {${\germ p}$}-adischen {K}örpern. {I}},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {88},
      year = {1965},
      pages = {40--47},
      issn = {0025-5874},
      mrclass = {14.50},
      mrnumber = {0174559},
      mrreviewer = {R. Steinberg},
      doi = {10.1007/BF01112691},
      url = {https://doi.org/10.1007/BF01112691},
      zblnumber = {0143.04702},
      }
  • [Kowalski] Go to document E. Kowalski, The Large Sieve and its Applications, Cambridge University Press, Cambridge, 2008, vol. 175.
    @BOOK{Kowalski,
      author = {Kowalski, E.},
      title = {The Large Sieve and its Applications},
      series = {Cambridge Tracts in Math.},
      volume = {175},
      titlenote = {Arithmetic Geometry, Random Walks and Discrete Groups},
      publisher = {Cambridge University Press, Cambridge},
      year = {2008},
      pages = {xxii+293},
      isbn = {978-0-521-88851-6},
      mrclass = {11N35 (11-02 11N36 60G99)},
      mrnumber = {2426239},
      mrreviewer = {Harald A. Helfgott},
      doi = {10.1017/CBO9780511542947},
      url = {https://doi.org/10.1017/CBO9780511542947},
      zblnumber = {1177.11080},
      }
  • [Landau1918] Go to document E. Landau, "Über Imaginär- quadratischer Zahlkörper," Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, vol. 1918, pp. 285-295, 1918.
    @ARTICLE{Landau1918,
      author = {Landau, E.},
      journal = {Nachrichten von der Gesellschaft der Wissenschaften zu G{ö}ttingen, Mathematisch-Physikalische Klasse},
      pages = {285-295},
      title = {{Ü}ber Imagin{ä}r- quadratischer Zahlk{ö}rper},
      url = {http://eudml.org/doc/59028},
      volume = {1918},
      year = {1918},
      zblnumber = {46.0258.04},
      }
  • [Lemmermeyer] Go to document F. Lemmermeyer, "The development of the principal genus theorem," in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Springer, Berlin, 2007, pp. 529-561.
    @INCOLLECTION{Lemmermeyer,
      author = {Lemmermeyer, Franz},
      title = {The development of the principal genus theorem},
      booktitle = {The Shaping of Arithmetic after {C}. {F}. {G}auss's {\it {D}isquisitiones Arithmeticae}},
      pages = {529--561},
      publisher = {Springer, Berlin},
      year = {2007},
      mrclass = {01A55 (11-03 11R29)},
      mrnumber = {2308295},
      doi = {10.1007/978-3-540-34720-0_20},
      url = {https://doi.org/10.1007/978-3-540-34720-0_20},
      zblnumber = {},
      }
  • [LinnikBook] Y. V. Linnik, Ergodic Properties of Algebraic Fields, Springer-Verlag New York Inc., New York, 1968.
    @BOOK{LinnikBook,
      author = {Linnik, Yu. V.},
      title = {Ergodic Properties of Algebraic Fields},
      series = {translated from the Russian by M. S. Keane, {\em Ergeb. Math. Grenzgeb.} {\bf 45}},
      publisher = {Springer-Verlag New York Inc., New York},
      year = {1968},
      pages = {ix+192},
      mrclass = {10.65},
      mrnumber = {0238801},
      zblnumber = {0162.06801},
      }
  • [QuantumLindenstrauss] Go to document E. Lindenstrauss, "Invariant measures and arithmetic quantum unique ergodicity," Ann. of Math. (2), vol. 163, iss. 1, pp. 165-219, 2006.
    @ARTICLE{QuantumLindenstrauss,
      author = {Lindenstrauss, Elon},
      title = {Invariant measures and arithmetic quantum unique ergodicity},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {163},
      year = {2006},
      number = {1},
      pages = {165--219},
      issn = {0003-486X},
      mrclass = {11F72 (37A45 37D40)},
      mrnumber = {2195133},
      mrreviewer = {Zeév Rudnick},
      doi = {10.4007/annals.2006.163.165},
      url = {https://doi.org/10.4007/annals.2006.163.165},
      zblnumber = {1104.22015},
      }
  • [LRS] W. Luo, Z. Rudnick, and P. Sarnak, "On the generalized Ramanujan conjecture for ${ GL}(n)$," in Automorphic Forms, Automorphic Representations, and Arithmetic, Amer. Math. Soc., Providence, RI, 1999, vol. 66, pp. 301-310.
    @INCOLLECTION{LRS,
      author = {Luo, Wenzhi and Rudnick, Zeév and Sarnak, Peter},
      title = {On the generalized {R}amanujan conjecture for {${\rm GL}(n)$}},
      booktitle = {Automorphic Forms, Automorphic Representations, and Arithmetic},
      venue = {{F}ort {W}orth, {TX},
      1996},
      series = {Proc. Sympos. Pure Math.},
      volume = {66},
      pages = {301--310},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1999},
      mrclass = {11F70 (11F72)},
      mrnumber = {1703764},
      mrreviewer = {Stefan Kühnlein},
      zblnumber = {0965.11023},
      }
  • [GIT] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Third ed., Springer-Verlag, Berlin, 1994, vol. 34.
    @BOOK{GIT,
      author = {Mumford, D. and Fogarty, J. and Kirwan, F.},
      title = {Geometric Invariant Theory},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {34},
      edition = {Third},
      publisher = {Springer-Verlag, Berlin},
      year = {1994},
      pages = {xiv+292},
      isbn = {3-540-56963-4},
      mrclass = {14D25 (58E05 58F05)},
      mrnumber = {1304906},
      mrreviewer = {Yi Hu},
      zblnumber = {0797.14004},
      }
  • [Michel04] Go to document P. Michel, "The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points," Ann. of Math. (2), vol. 160, iss. 1, pp. 185-236, 2004.
    @ARTICLE{Michel04,
      author = {Michel, P.},
      title = {The subconvexity problem for {R}ankin-{S}elberg {$L$}-functions and equidistribution of {H}eegner points},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {160},
      year = {2004},
      number = {1},
      pages = {185--236},
      issn = {0003-486X},
      mrclass = {11F66 (11G18)},
      mrnumber = {2119720},
      mrreviewer = {Gergely Harcos},
      doi = {10.4007/annals.2004.160.185},
      url = {https://doi.org/10.4007/annals.2004.160.185},
      zblnumber = {1068.11033},
      }
  • [Milne] Go to document J. S. Milne, "Introduction to Shimura varieties," in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Amer. Math. Soc., Providence, RI, 2005, vol. 4, pp. 265-378.
    @INCOLLECTION{Milne,
      author = {Milne, J. S.},
      title = {Introduction to {S}himura varieties},
      booktitle = {Harmonic Analysis, the Trace Formula, and {S}himura Varieties},
      series = {Clay Math. Proc.},
      volume = {4},
      pages = {265--378},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2005},
      mrclass = {11G18 (14G35)},
      mrnumber = {2192012},
      mrreviewer = {Ulrich Görtz},
      zblnumber = {1148.14011},
      url = {http://www.claymath.org/library/proceedings/cmip04.pdf},
      }
  • [MargulisTomanov] Go to document G. A. Margulis and G. M. Tomanov, "Invariant measures for actions of unipotent groups over local fields on homogeneous spaces," Invent. Math., vol. 116, iss. 1-3, pp. 347-392, 1994.
    @ARTICLE{MargulisTomanov,
      author = {Margulis, G. A. and Tomanov, G. M.},
      title = {Invariant measures for actions of unipotent groups over local fields on homogeneous spaces},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {116},
      year = {1994},
      number = {1-3},
      pages = {347--392},
      issn = {0020-9910},
      mrclass = {22E40 (11E57 20G25 22D40)},
      mrnumber = {1253197},
      mrreviewer = {Nimish A. Shah},
      doi = {10.1007/BF01231565},
      url = {https://doi.org/10.1007/BF01231565},
      zblnumber = {0816.22004},
      }
  • [MichelVenkateshRev] P. Michel and A. Venkatesh, "Equidistribution, $L$-functions and ergodic theory: on some problems of Yu. Linnik," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 421-457.
    @INCOLLECTION{MichelVenkateshRev,
      author = {Michel, Philippe and Venkatesh, Akshay},
      title = {Equidistribution, {$L$}-functions and ergodic theory: on some problems of {Y}u. {L}innik},
      booktitle = {International {C}ongress of {M}athematicians. {V}ol. {II}},
      pages = {421--457},
      publisher = {Eur. Math. Soc., Zürich},
      year = {2006},
      mrclass = {11F66 (11F67 11M41 37A45)},
      mrnumber = {2275604},
      mrreviewer = {Gergely Harcos},
      zblnumber = {1157.11019},
      }
  • [Nagata] Go to document M. Nagata, "Invariants of a group in an affine ring," J. Math. Kyoto Univ., vol. 3, pp. 369-377, 1963/1964.
    @ARTICLE{Nagata,
      author = {Nagata, Masayoshi},
      title = {Invariants of a group in an affine ring},
      journal = {J. Math. Kyoto Univ.},
      fjournal = {Journal of Mathematics of Kyoto University},
      volume = {3},
      year = {1963/1964},
      pages = {369--377},
      issn = {0023-608X},
      mrclass = {13.93 (14.05)},
      mrnumber = {0179268},
      mrreviewer = {G. E. Bredon},
      doi = {10.1215/kjm/1250524787},
      url = {https://doi.org/10.1215/kjm/1250524787},
      zblnumber = {0146.04501},
      }
  • [Nair] Go to document M. Nair, "Multiplicative functions of polynomial values in short intervals," Acta Arith., vol. 62, iss. 3, pp. 257-269, 1992.
    @ARTICLE{Nair,
      author = {Nair, Mohan},
      title = {Multiplicative functions of polynomial values in short intervals},
      journal = {Acta Arith.},
      fjournal = {Acta Arithmetica},
      volume = {62},
      year = {1992},
      number = {3},
      pages = {257--269},
      issn = {0065-1036},
      mrclass = {11N32 (11N64)},
      mrnumber = {1197420},
      mrreviewer = {Michael Filaseta},
      doi = {10.4064/aa-62-3-257-269},
      url = {https://doi.org/10.4064/aa-62-3-257-269},
      zblnumber = {0768.11038},
      }
  • [NairTenenbaum] Go to document M. Nair and G. Tenenbaum, "Short sums of certain arithmetic functions," Acta Math., vol. 180, iss. 1, pp. 119-144, 1998.
    @ARTICLE{NairTenenbaum,
      author = {Nair, Mohan and Tenenbaum, Gérald},
      title = {Short sums of certain arithmetic functions},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {180},
      year = {1998},
      number = {1},
      pages = {119--144},
      issn = {0001-5962},
      mrclass = {11N37 (11N25 11N32)},
      mrnumber = {1618321},
      mrreviewer = {A. J. Hildebrand},
      doi = {10.1007/BF02392880},
      url = {https://doi.org/10.1007/BF02392880},
      zblnumber = {0917.11048},
      }
  • [Pila] Go to document J. Pila, "O-minimality and the André-Oort conjecture for $\Bbb C^n$," Ann. of Math. (2), vol. 173, iss. 3, pp. 1779-1840, 2011.
    @ARTICLE{Pila,
      author = {Pila, Jonathan},
      title = {O-minimality and the {A}ndré-{O}ort conjecture for {$\Bbb C^n$}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {173},
      year = {2011},
      number = {3},
      pages = {1779--1840},
      issn = {0003-486X},
      mrclass = {11G18 (03C64 11U09 14G35)},
      mrnumber = {2800724},
      mrreviewer = {Alexandra Shlapentokh},
      doi = {10.4007/annals.2011.173.3.11},
      url = {https://doi.org/10.4007/annals.2011.173.3.11},
      zblnumber = {1243.14022},
      }
  • [PilaTsimerman] Go to document J. Pila and J. Tsimerman, "The André-Oort conjecture for the moduli space of abelian surfaces," Compos. Math., vol. 149, iss. 2, pp. 204-216, 2013.
    @ARTICLE{PilaTsimerman,
      author = {Pila, Jonathan and Tsimerman, Jacob},
      title = {The {A}ndré-{O}ort conjecture for the moduli space of abelian surfaces},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {149},
      year = {2013},
      number = {2},
      pages = {204--216},
      issn = {0010-437X},
      mrclass = {11G18 (03C64 14K10)},
      mrnumber = {3020307},
      mrreviewer = {Jae-Hyun Yang},
      doi = {10.1112/S0010437X12000589},
      url = {https://doi.org/10.1112/S0010437X12000589},
      zblnumber = {1304.11055},
      }
  • [Reiner] I. Reiner, Maximal Orders, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1975, vol. 5.
    @BOOK{Reiner,
      author = {Reiner, I.},
      title = {Maximal Orders},
      series = {London Math. Soc. Monogr.},
      volume = {5},
      publisher = {Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York},
      year = {1975},
      pages = {xii+395},
      mrclass = {16A18 (12A80)},
      mrnumber = {0393100},
      mrreviewer = {H. Jacobinski},
      zblnumber = {1024.16008},
      }
  • [Sarnak] P. C. Sarnak, "Diophantine problems and linear groups," in Proceedings of the International Congress of Mathematicians, Vol. I, II, 1991, pp. 459-471.
    @INPROCEEDINGS{Sarnak,
      author = {Sarnak, Peter C.},
      title = {Diophantine problems and linear groups},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {I},
      {II}},
      venue = {{K}yoto, 1990},
      pages = {459--471},
      publisher = {Math. Soc. Japan, Tokyo},
      year = {1991},
      mrclass = {11F72 (11D99 22E40)},
      mrnumber = {1159234},
      mrreviewer = {Jonathan Huntley},
      zblnumber = {0743.11018},
      }
  • [Schwartz] Go to document J. T. Schwartz, "Fast probabilistic algorithms for verification of polynomial identities," J. Assoc. Comput. Mach., vol. 27, iss. 4, pp. 701-717, 1980.
    @ARTICLE{Schwartz,
      author = {Schwartz, J. T.},
      title = {Fast probabilistic algorithms for verification of polynomial identities},
      journal = {J. Assoc. Comput. Mach.},
      fjournal = {Journal of the Association for Computing Machinery},
      volume = {27},
      year = {1980},
      number = {4},
      pages = {701--717},
      issn = {0004-5411},
      mrclass = {68C20 (03B35 12-04)},
      mrnumber = {0594695},
      doi = {10.1145/322217.322225},
      url = {https://doi.org/10.1145/322217.322225},
      zblnumber = {0452.68050},
      }
  • [Shahidi] Go to document F. Shahidi, "On the Ramanujan conjecture and finiteness of poles for certain $L$-functions," Ann. of Math. (2), vol. 127, iss. 3, pp. 547-584, 1988.
    @ARTICLE{Shahidi,
      author = {Shahidi, Freydoon},
      title = {On the {R}amanujan conjecture and finiteness of poles for certain {$L$}-functions},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {127},
      year = {1988},
      number = {3},
      pages = {547--584},
      issn = {0003-486X},
      mrclass = {11F70 (11F66 11F67 22E50 22E55)},
      mrnumber = {0942520},
      mrreviewer = {David Joyner},
      doi = {10.2307/2007005},
      url = {https://doi.org/10.2307/2007005},
      zblnumber = {0654.10029},
      }
  • [Shiu] Go to document P. Shiu, "A Brun-Titchmarsh theorem for multiplicative functions," J. Reine Angew. Math., vol. 313, pp. 161-170, 1980.
    @ARTICLE{Shiu,
      author = {Shiu, P.},
      title = {A {B}run-{T}itchmarsh theorem for multiplicative functions},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik},
      volume = {313},
      year = {1980},
      pages = {161--170},
      issn = {0075-4102},
      mrclass = {10H25},
      mrnumber = {0552470},
      mrreviewer = {A. I. Vinogradov},
      doi = {10.1515/crll.1980.313.161},
      url = {https://doi.org/10.1515/crll.1980.313.161},
      zblnumber = {0412.10030},
      }
  • [ShendeTsimerman] Go to document V. Shende and J. Tsimerman, "Equidistribution in ${ Bun}_2(\Bbb{P}^1)$," Duke Math. J., vol. 166, iss. 18, pp. 3461-3504, 2017.
    @ARTICLE{ShendeTsimerman,
      author = {Shende, Vivek and Tsimerman, Jacob},
      title = {Equidistribution in {${\rm Bun}_2(\Bbb{P}^1)$}},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {166},
      year = {2017},
      number = {18},
      pages = {3461--3504},
      issn = {0012-7094},
      mrclass = {11G20 (14H40 14H51)},
      mrnumber = {3732881},
      mrreviewer = {Martha Rzedowski-Calderón},
      doi = {10.1215/00127094-2017-0025},
      url = {https://doi.org/10.1215/00127094-2017-0025},
      zblnumber = {06837465},
      }
  • [TaoPolynomial] Go to document T. Tao, "Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory," EMS Surv. Math. Sci., vol. 1, iss. 1, pp. 1-46, 2014.
    @ARTICLE{TaoPolynomial,
      author = {Tao, Terence},
      title = {Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory},
      journal = {EMS Surv. Math. Sci.},
      fjournal = {EMS Surveys in Mathematical Sciences},
      volume = {1},
      year = {2014},
      number = {1},
      pages = {1--46},
      issn = {2308-2151},
      mrclass = {14-02 (05-02 11B30 14G15 14Q99 51E30)},
      mrnumber = {3200226},
      mrreviewer = {Victoria Powers},
      doi = {10.4171/EMSS/1},
      url = {https://doi.org/10.4171/EMSS/1},
      zblnumber = {1294.05044},
      }
  • [Tsimerman] Go to document J. Tsimerman, "The André-Oort conjecture for $\mathcal{A}_g$," Ann. of Math. (2), vol. 187, iss. 2, pp. 379-390, 2018.
    @ARTICLE{Tsimerman,
      author = {Tsimerman, Jacob},
      title = {The {A}ndré-{O}ort conjecture for {$\mathcal{A}_g$}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {187},
      year = {2018},
      number = {2},
      pages = {379--390},
      issn = {0003-486X},
      mrclass = {11G15 (11G18 14G35)},
      mrnumber = {3744855},
      mrreviewer = {Patrick Morton},
      doi = {10.4007/annals.2018.187.2.2},
      url = {https://doi.org/10.4007/annals.2018.187.2.2},
      zblnumber = {06841543},
      }
  • [vanDerCorput] Go to document J. G. van der Corput, "Über Gitterpunkte in der Ebene," Math. Ann., vol. 81, iss. 1, pp. 1-20, 1920.
    @ARTICLE{vanDerCorput,
      author = {van der Corput, J. G.},
      title = {Über {G}itterpunkte in der {E}bene},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {81},
      year = {1920},
      number = {1},
      pages = {1--20},
      issn = {0025-5831},
      mrclass = {DML},
      mrnumber = {1511951},
      doi = {10.1007/BF01563613},
      url = {https://doi.org/10.1007/BF01563613},
      jfmnumber = {47.0159.01},
      }
  • [Zhang] Go to document S. Zhang, "Equidistribution of CM-points on quaternion Shimura varieties," Int. Math. Res. Not., iss. 59, pp. 3657-3689, 2005.
    @ARTICLE{Zhang,
      author = {Zhang, Shou-Wu},
      title = {Equidistribution of {CM}-points on quaternion {S}himura varieties},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      year = {2005},
      number = {59},
      pages = {3657--3689},
      issn = {1073-7928},
      mrclass = {11G18 (11F60 14G35)},
      mrnumber = {2200081},
      mrreviewer = {Akshay Venkatesh},
      doi = {10.1155/IMRN.2005.3657},
      url = {https://doi.org/10.1155/IMRN.2005.3657},
      zblnumber = {1096.14016},
      }
  • [Zippel] Go to document R. Zippel, "Probabilistic algorithms for sparse polynomials," in Symbolic and Algebraic Computation, Springer, Berlin-New York, 1979, vol. 72, pp. 216-226.
    @INCOLLECTION{Zippel,
      author = {Zippel, Richard},
      title = {Probabilistic algorithms for sparse polynomials},
      booktitle = {Symbolic and Algebraic Computation},
      venue = {{EUROSAM} '79, {I}nternat. {S}ympos., {M}arseille, 1979},
      series = {Lecture Notes in Comput. Sci.},
      volume = {72},
      pages = {216--226},
      publisher = {Springer, Berlin-New York},
      year = {1979},
      mrclass = {68C20},
      mrnumber = {0575692},
      zblnumber = {0418.68040},
      doi = {10.1007/3-540-09519-5_73},
      }

Authors

Ilya Khayutin

Princeton University, Princeton, NJ and Institute for Advanced Study, Princeton, NJ