Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Abstract

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let $\ell > 2$ be prime and $A$ a finite abelian $\ell$-group. Then there exists $Q = Q(A)$ such that, for $q$ greater than $Q$, a positive fraction of quadratic extensions of $\mathbb{F}_q(t)$ have the $\ell$-part of their class group isomorphic to $A$.

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  • [SGA7] Go to document Groupes de Monodromie en Géométrie Algébrique. II, Deligne, P. and Katz, N., Eds., New York: Springer-Verlag, 1973, vol. 340.
    @book{SGA7, mrkey = {0354657},
      title = {Groupes de Monodromie en Géométrie Algébrique. {II}},
      series = {Lecture Notes in Math.},
      volume = {340},
      note = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie 1967--1969 (SGA 7 II)},
      EDITOR={Deligne, Pierre and Katz, N.},
      publisher = {Springer-Verlag},
      year = {1973},
      pages = {x+438},
      mrclass = {14-06},
      mrnumber = {0354657},
      address = {New York},
      zblnumber = {0258.00005},
      doi = {10.1007/BFb0060505},
      }
  • [acv] Go to document D. Abramovich, A. Corti, and A. Vistoli, "Twisted bundles and admissible covers," Comm. Algebra, vol. 31, iss. 8, pp. 3547-3618, 2003.
    @article{acv, mrkey = {2007376},
      author = {Abramovich, Dan and Corti, Alessio and Vistoli, Angelo},
      title = {Twisted bundles and admissible covers},
      note = {special issue in honor of Steven L. Kleiman},
      journal = {Comm. Algebra},
      fjournal = {Communications in Algebra},
      volume = {31},
      year = {2003},
      number = {8},
      pages = {3547--3618},
      issn = {0092-7872},
      coden = {COALDM},
      mrclass = {14H10 (14A20 14H30)},
      mrnumber = {2007376},
      mrreviewer = {Andrew Kresch},
      doi = {10.1081/AGB-120022434},
      zblnumber = {1077.14034},
      }
  • [Achter] Go to document J. D. Achter, "Results of Cohen-Lenstra type for quadratic function fields," in Computational Arithmetic Geometry, Providence, RI: Amer. Math. Soc., 2008, vol. 463, pp. 1-7.
    @incollection{Achter, mrkey = {2459984},
      author = {Achter, Jeffrey D.},
      title = {Results of {C}ohen-{L}enstra type for quadratic function fields},
      booktitle = {Computational Arithmetic Geometry},
      series = {Contemp. Math.},
      volume = {463},
      pages = {1--7},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2008},
      mrclass = {11G20 (11R58 14G15)},
      mrnumber = {2459984},
      mrreviewer = {Ernst-Ulrich Gekeler},
      doi = {10.1090/conm/463/09041},
      zblnumber = {1166.11018},
      }
  • [achterpries] Go to document J. D. Achter and R. Pries, "The integral monodromy of hyperelliptic and trielliptic curves," Math. Ann., vol. 338, iss. 1, pp. 187-206, 2007.
    @article{achterpries, mrkey = {2295509},
      author = {Achter, Jeffrey D. and Pries, Rachel},
      title = {The integral monodromy of hyperelliptic and trielliptic curves},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {338},
      year = {2007},
      number = {1},
      pages = {187--206},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {11G18 (14D05)},
      mrnumber = {2295509},
      mrreviewer = {Gerhard Pfister},
      doi = {10.1007/s00208-006-0072-0},
      zblnumber = {1129.11027},
      }
  • [Arnold] Go to document V. I. Arnolcprimed, "The cohomology ring of the group of dyed braids," Mat. Zametki, vol. 5, pp. 227-231, 1969.
    @article{Arnold, mrkey = {0242196},
      author = {Arnol{\cprime}d, V. I.},
      title = {The cohomology ring of the group of dyed braids},
      journal = {Mat. Zametki},
      fjournal = {Akademiya Nauk SSSR. Matematicheskie Zametki},
      volume = {5},
      year = {1969},
      pages = {227--231},
      issn = {0025-567X},
      mrclass = {57.60},
      mrnumber = {0242196},
      mrreviewer = {C. Teleman},
      zblnumber = {0277.55002},
      doi = {10.1007/BF01098313},
      }
  • [Artin] Go to document E. Artin, "Theory of braids," Ann. of Math., vol. 48, pp. 101-126, 1947.
    @article{Artin, mrkey = {0019087},
      author = {Artin, E.},
      title = {Theory of braids},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {48},
      year = {1947},
      pages = {101--126},
      issn = {0003-486X},
      mrclass = {20.0X},
      mrnumber = {0019087},
      mrreviewer = {S. Eilenberg},
      doi = {10.2307/1969218},
      zblnumber = {0030.17703},
      }
  • [Borel] Go to document A. Borel, "Stable real cohomology of arithmetic groups," Ann. Sci. École Norm. Sup., vol. 7, pp. 235-272 (1975), 1974.
    @article{Borel, mrkey = {0387496},
      author = {Borel, Armand},
      title = {Stable real cohomology of arithmetic groups},
      journal = {Ann. Sci. École Norm. Sup.},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      volume = {7},
      year = {1974},
      pages = {235--272 (1975)},
      issn = {0012-9593},
      mrclass = {22E40 (20G10)},
      mrnumber = {0387496},
      mrreviewer = {Howard Garland},
      url = {http://www.numdam.org/item?id=ASENS_1974_4_7_2_235_0},
      zblnumber = {0316.57026},
      }
  • [Milgram] Go to document C. P. Boyer, J. C. Hurtubise, and R. J. Milgram, "Stability theorems for spaces of rational curves," Internat. J. Math., vol. 12, iss. 2, pp. 223-262, 2001.
    @article{Milgram, mrkey = {1823576},
      author = {Boyer, C. P. and Hurtubise, J. C. and Milgram, R. J.},
      title = {Stability theorems for spaces of rational curves},
      journal = {Internat. J. Math.},
      fjournal = {International Journal of Mathematics},
      volume = {12},
      year = {2001},
      number = {2},
      pages = {223--262},
      issn = {0129-167X},
      mrclass = {55P10 (32C18 58D15)},
      mrnumber = {1823576},
      mrreviewer = {T. Arleigh Crawford},
      doi = {10.1142/S0129167X01000721},
      zblnumber = {1110.58303},
      }
  • [byeo:indivisibility] Go to document D. Byeon, "Indivisibility of class numbers of imaginary quadratic function fields," Acta Arith., vol. 132, iss. 4, pp. 373-376, 2008.
    @article{byeo:indivisibility, mrkey = {2413359},
      author = {Byeon, Dongho},
      title = {Indivisibility of class numbers of imaginary quadratic function fields},
      journal = {Acta Arith.},
      fjournal = {Acta Arithmetica},
      volume = {132},
      year = {2008},
      number = {4},
      pages = {373--376},
      issn = {0065-1036},
      coden = {AARIA9},
      mrclass = {11R58 (11R29)},
      mrnumber = {2413359},
      mrreviewer = {Akira Aiba},
      doi = {10.4064/aa132-4-6},
      zblnumber = {1170.11042},
      }
  • [Carlitz] Go to document L. Carlitz, "The arithmetic of polynomials in a Galois field," Proc. Natl. Acad. Sci. USA, vol. 17, pp. 120-122, 1931.
    @article{Carlitz,
      author = {Carlitz, L},
      title = {The arithmetic of polynomials in a {G}alois field},
      journal = {Proc. Natl. Acad. Sci. USA},
      volume = {17},
      year = {1931},
      pages = {120--122},
      zblnumber = {0001.12401},
      url = {http://www.pnas.org/content/17/2/120.full.pdf},
      }
  • [Charney-Davis] R. Charney and M. W. Davis, "Finite $K(\pi, 1)$s for Artin groups," in Prospects in Topology, Princeton, NJ: Princeton Univ. Press, 1995, vol. 138, pp. 110-124.
    @incollection{Charney-Davis, mrkey = {1368655},
      author = {Charney, Ruth and Davis, Michael W.},
      title = {Finite {$K(\pi, 1)$}s for {A}rtin groups},
      booktitle = {Prospects in Topology},
      venue = {{P}rinceton, {NJ},
      1994},
      series = {Ann. of Math. Stud.},
      volume = {138},
      pages = {110--124},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {1995},
      mrclass = {57M07 (20F36 55P20)},
      mrnumber = {1368655},
      mrreviewer = {Darryl McCullough},
      zblnumber = {0930.55006},
      }
  • [Cohen] Go to document F. R. Cohen, T. J. Lada, and P. J. May, The Homology of Iterated Loop Spaces, New York: Springer-Verlag, 1976, vol. 533.
    @book{Cohen, mrkey = {0436146},
      author = {Cohen, Frederick R. and Lada, Thomas J. and May, J. Peter},
      title = {The Homology of Iterated Loop Spaces},
      series = {Lecture Notes in Math.},
      volume = {533},
      publisher = {Springer-Verlag},
      year = {1976},
      pages = {vii+490},
      mrclass = {55G25 (55D35)},
      mrnumber = {0436146},
      mrreviewer = {Peter J. Eccles},
      address = {New York},
      zblnumber = {0334.55009},
      doi = {10.1007/BFb0080464},
      }
  • [cohe:cohenlenstra] Go to document H. Cohen and H. W. Lenstra Jr., "Heuristics on class groups of number fields," in Number theory, Noordwijkerhout 1983, New York: Springer-Verlag, 1984, vol. 1068, pp. 33-62.
    @incollection{cohe:cohenlenstra, mrkey = {0756082},
      author = {Cohen, H. and Lenstra, Jr., H. W.},
      title = {Heuristics on class groups of number fields},
      booktitle = {Number theory, {N}oordwijkerhout 1983},
      venue = {{N}oordwijkerhout, 1983},
      series = {Lecture Notes in Math.},
      volume = {1068},
      pages = {33--62},
      publisher = {Springer-Verlag},
      year = {1984},
      mrclass = {11R29 (13C13)},
      mrnumber = {0756082},
      mrreviewer = {F. J. van der Linden},
      doi = {10.1007/BFb0099440},
      address = {New York},
      zblnumber = {0558.12002},
      }
  • [cohenmartinet] Go to document H. Cohen and J. Martinet, "Étude heuristique des groupes de classes des corps de nombres," J. Reine Angew. Math., vol. 404, pp. 39-76, 1990.
    @article{cohenmartinet, mrkey = {1037430},
      author = {Cohen, Henri and Martinet, Jacques},
      title = {\'{E}tude heuristique des groupes de classes des corps de nombres},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik},
      volume = {404},
      year = {1990},
      pages = {39--76},
      issn = {0075-4102},
      coden = {JRMAA8},
      mrclass = {11R29},
      mrnumber = {1037430},
      zblnumber = {0699.12016},
      doi = {10.1515/crll.1990.404.39},
      }
  • [Damiolini] Go to document C. Damiolini, The braid group and the arc complex, 2013.
    @misc{Damiolini,
      author = {Damiolini, Chiara},
      title = {The braid group and the arc complex},
      url = {http://www.math.leidenuniv.nl/scripties/DamioliniMaster.pdf?8},
      year = {2013},
      }
  • [dats:dawr] Go to document B. Datskovsky and D. J. Wright, "Density of discriminants of cubic extensions," J. Reine Angew. Math., vol. 386, pp. 116-138, 1988.
    @article{dats:dawr, mrkey = {0936994},
      author = {Datskovsky, Boris and Wright, David J.},
      title = {Density of discriminants of cubic extensions},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik},
      volume = {386},
      year = {1988},
      pages = {116--138},
      issn = {0075-4102},
      coden = {JRMAA8},
      mrclass = {11R29 (11R11 11R16)},
      mrnumber = {0936994},
      mrreviewer = {Brian Peterson},
      doi = {10.1515/crll.1988.386.116},
      zblnumber = {0632.12007},
      }
  • [pace:pacellireu] Go to document M. Daub, J. Lang, M. Merling, A. M. Pacelli, N. Pitiwan, and M. Rosen, Function fields with class number indivisible by a prime $\ell$, 2011.
    @misc{pace:pacellireu, MRKEY={2847264},
      AUTHOR = {Daub, Michael and Lang, Jaclyn and Merling, Mona and Pacelli, Allison M. and Pitiwan, Natee and Rosen, Michael},
      TITLE = {Function fields with class number indivisible by a prime {$\ell$}},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {150},
      YEAR = {2011},
      NUMBER = {4},
      PAGES = {339--359},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11R29 (11R58)},
      MRNUMBER = {2847264},
      MRREVIEWER = {J{ü}rgen Ritter},
      zblnumber = {1263.11098},
      DOI = {10.4064/aa150-4-2},
     }
  • [dejongkatz] A. J. DeJong and N. Katz, Counting the number of curves over a finite field, 2001.
    @misc{dejongkatz,
      author = {DeJong, A. J. and Katz, N.},
      title = {Counting the number of curves over a finite field},
      year = {2001},
      note = {preprint},
      }
  • [DR] Go to document P. Deligne and M. Rapoport, "Les schémas de modules de courbes elliptiques," in Modular Functions of One Variable, II, New York: Springer-Verlag, 1973, vol. 349, pp. 143-316.
    @incollection{DR, mrkey = {0337993},
      author = {Deligne, Pierre and Rapoport, M.},
      title = {Les schémas de modules de courbes elliptiques},
      booktitle = {Modular Functions of One Variable, {II}},
      venue = {{P}roc. {I}nternat. {S}ummer {S}chool, {U}niv. {A}ntwerp, {A}ntwerp, 1972},
      pages = {143--316},
      series = {Lecture Notes in Math.},
      volume = {349},
      publisher = {Springer-Verlag},
      year = {1973},
      mrclass = {14K10 (10D05)},
      mrnumber = {0337993},
      mrreviewer = {T. Oda},
      address = {New York},
      zblnumber = {0281.14010},
      doi = {10.1007/978-3-540-37855-6},
      }
  • [DeligneWeil] Go to document P. Deligne, "La conjecture de Weil. I," Inst. Hautes Études Sci. Publ. Math., vol. 43, pp. 273-307, 1974.
    @article{DeligneWeil, mrkey = {0340258},
      author = {Deligne, Pierre},
      title = {La conjecture de {W}eil. {I}},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      volume = {43},
      year = {1974},
      pages = {273--307},
      issn = {0073-8301},
      mrclass = {14G13},
      mrnumber = {0340258},
      mrreviewer = {Nicholas M. Katz},
      doi = {10.1007/BF02684373},
      zblnumber={0287.14001},
      }
  • [Deligne] Go to document P. Deligne, "La conjecture de Weil. II," Inst. Hautes Études Sci. Publ. Math., vol. 52, pp. 137-252, 1980.
    @article{Deligne, mrkey = {0601520},
      author = {Deligne, Pierre},
      title = {La conjecture de {W}eil. {II}},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      volume = {52},
      year = {1980},
      pages = {137--252},
      issn = {0073-8301},
      coden = {PMIHA6},
      mrclass = {14G13 (10H10)},
      mrnumber = {0601520},
      mrreviewer = {Spencer J. Bloch},
      doi = {10.1007/BF02684780},
      zblnumber = {0456.14014},
      }
  • [EV] Go to document J. S. Ellenberg and A. Venkatesh, "Counting extensions of function fields with bounded discriminant and specified Galois group," in Geometric Methods in Algebra and Number Theory, Boston: Birkhäuser, 2005, vol. 235, pp. 151-168.
    @incollection{EV, mrkey = {2159381},
      author = {Ellenberg, Jordan S. and Venkatesh, Akshay},
      title = {Counting extensions of function fields with bounded discriminant and specified {G}alois group},
      booktitle = {Geometric Methods in Algebra and Number Theory},
      series = {Progr. Math.},
      volume = {235},
      pages = {151--168},
      publisher = {Birkhäuser},
      address = {Boston},
      year = {2005},
      mrclass = {11R58 (12F12)},
      mrnumber = {2159381},
      mrreviewer = {Xianke Zhang},
      doi = {10.1007/0-8176-4417-2_7},
      zblnumber = {1085.11057},
      }
  • [LettertodeJong] J. S. Ellenberg and A. Venkatesh, Letter to A. J. de Jong, 2008.
    @misc{LettertodeJong,
      author = {Ellenberg, Jordan S. and Venkatesh, Akshay},
      title = {Letter to {A}. {J}. de {J}ong},
      year = {2008},
      }
  • [fouv:4rank] Go to document &. Fouvry and J. Klüners, "On the 4-rank of class groups of quadratic number fields," Invent. Math., vol. 167, iss. 3, pp. 455-513, 2007.
    @article{fouv:4rank, mrkey = {2276261},
      author = {Fouvry, {É}tienne and Kl{ü}ners, J{ü}rgen},
      title = {On the 4-rank of class groups of quadratic number fields},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {167},
      year = {2007},
      number = {3},
      pages = {455--513},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {11R29 (11R11 11R45)},
      mrnumber = {2276261},
      mrreviewer = {J. Browkin},
      doi = {10.1007/s00222-006-0021-2},
      zblnumber = {1126.11062},
     }
  • [FK] Go to document E. Freitag and R. Kiehl, Étale Cohomology and the Weil Conjecture, New York: Springer-Verlag, 1988, vol. 13.
    @book{FK, mrkey = {0926276},
      author = {Freitag, Eberhard and Kiehl, Reinhardt},
      title = {\'{E}tale Cohomology and the {W}eil Conjecture},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {13},
      note = {translated from the German by Betty S. Waterhouse and William C. Waterhouse, with an historical introduction by J. A. Dieudonn{é}},
      publisher = {Springer-Verlag},
      year = {1988},
      pages = {xviii+317},
      isbn = {3-540-12175-7},
      mrclass = {14F20 (11G25 14G13)},
      mrnumber = {0926276},
      mrreviewer = {James Milne},
      doi = {10.1007/978-3-662-02541-3},
      address = {New York},
      zblnumber = {0643.14012},
     }
  • [friedvolklein] Go to document M. D. Fried and H. Völklein, "The inverse Galois problem and rational points on moduli spaces," Math. Ann., vol. 290, iss. 4, pp. 771-800, 1991.
    @article{friedvolklein, mrkey = {1119950},
      author = {Fried, Michael D. and V{ö}lklein, Helmut},
      title = {The inverse {G}alois problem and rational points on moduli spaces},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {290},
      year = {1991},
      number = {4},
      pages = {771--800},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {12F12 (12L10 14E22 14H10 14H30)},
      mrnumber = {1119950},
      mrreviewer = {B. Heinrich Matzat},
      doi = {10.1007/BF01459271},
      zblnumber = {0763.12004},
     }
  • [FW] Go to document E. Friedman and L. C. Washington, "On the distribution of divisor class groups of curves over a finite field," in Théorie des Nombres, Berlin: de Gruyter, 1989, pp. 227-239.
    @incollection{FW, mrkey = {1024565},
      author = {Friedman, Eduardo and Washington, Lawrence C.},
      title = {On the distribution of divisor class groups of curves over a finite field},
      booktitle = {Théorie des Nombres},
      venue = {{Q}uebec, {PQ},
      1987},
      pages = {227--239},
      publisher = {de Gruyter},
      address = {Berlin},
      year = {1989},
      mrclass = {11R58 (11R29)},
      mrnumber = {1024565},
      mrreviewer = {H.-G. R{ü}ck},
      doi = {10.1515/9783110852790.227},
      zblnumber = {0693.12013},
     }
  • [frie:divisibility] Go to document C. Friesen, "Class number divisibility in real quadratic function fields," Canad. Math. Bull., vol. 35, iss. 3, pp. 361-370, 1992.
    @article{frie:divisibility, mrkey = {1184013},
      author = {Friesen, Christian},
      title = {Class number divisibility in real quadratic function fields},
      journal = {Canad. Math. Bull.},
      fjournal = {Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques},
      volume = {35},
      year = {1992},
      number = {3},
      pages = {361--370},
      issn = {0008-4395},
      coden = {CMBUA3},
      mrclass = {11R58 (11R29)},
      mrnumber = {1184013},
      mrreviewer = {Amara H{é}di},
      doi = {10.4153/CMB-1992-048-5},
      zblnumber = {0727.11045},
      }
  • [gartonthesis] Go to document D. Garton, "Random matrices, the Cohen-Lenstra heuristics, and roots of unity," Algebra Number Theory, vol. 9, iss. 1, pp. 149-171, 2015.
    @article{gartonthesis, mrkey = {3317763},
      author = {Garton, Derek},
      title = {Random matrices, the {C}ohen-{L}enstra heuristics, and roots of unity},
      journal = {Algebra Number Theory},
      fjournal = {Algebra \& Number Theory},
      volume = {9},
      year = {2015},
      number = {1},
      pages = {149--171},
      issn = {1937-0652},
      mrclass = {11R29 (11R58 15B52)},
      mrnumber = {3317763},
      mrreviewer = {Steven Joel Miller},
      doi = {10.2140/ant.2015.9.149},
      zblnumber = {06424744},
     }
  • [SGA1] A. Grothendieck, Revêtements étales et Groupe Fondamental. Fasc. II: Exposés 6, 8 à 11, Paris: Institut des Hautes Études Scientifiques, 1963, vol. 1960/61.
    @book{SGA1,
      author = {Grothendieck, Alexander},
      title = {Revêtements étales et Groupe Fondamental. {F}asc. {II}: {E}xposés 6, 8 à 11},
      series = {Séminaire de Géométrie Algébrique},
      volume = {1960/61},
      publisher = {Institut des Hautes Études Scientifiques},
      address = {Paris},
      year = {1963},
      pages = {i+163 pp. (not consecutively paged) (loose errata)},
      mrclass = {14.55},
      mrnumber = {0217088},
      mrreviewer = {T. Oda},
      }
  • [hainlooijenga] R. Hain and E. Looijenga, "Mapping class groups and moduli spaces of curves," in Algebraic Geometry—Santa Cruz 1995, Providence, RI: Amer. Math. Soc., 1997, vol. 62, pp. 97-142.
    @incollection{hainlooijenga, mrkey = {1492535},
      author = {Hain, Richard and Looijenga, Eduard},
      title = {Mapping class groups and moduli spaces of curves},
      booktitle = {Algebraic Geometry---{S}anta {C}ruz 1995},
      series = {Proc. Sympos. Pure Math.},
      volume = {62},
      pages = {97--142},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1997},
      mrclass = {14H10 (14C30 14H15 32G15 32G20)},
      mrnumber = {1492535},
      mrreviewer = {Takashi Ichikawa},
      zblnumber = {0914.14013},
     }
  • [cjh] Go to document C. Hall, "Big symplectic or orthogonal monodromy modulo $l$," Duke Math. J., vol. 141, iss. 1, pp. 179-203, 2008.
    @article{cjh, mrkey = {2372151},
      author = {Hall, Chris},
      title = {Big symplectic or orthogonal monodromy modulo {$l$}},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {141},
      year = {2008},
      number = {1},
      pages = {179--203},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {11G05 (11G10 12F12 14D05 14K15)},
      mrnumber = {2372151},
      mrreviewer = {Jeffrey D. Achter},
      doi = {10.1215/S0012-7094-08-14115-8},
      zblnumber = {1205.11062},
     }
  • [Harer] Go to document J. L. Harer, "Stability of the homology of the mapping class groups of orientable surfaces," Ann. of Math., vol. 121, iss. 2, pp. 215-249, 1985.
    @article{Harer, mrkey = {0786348},
      author = {Harer, John L.},
      title = {Stability of the homology of the mapping class groups of orientable surfaces},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {121},
      year = {1985},
      number = {2},
      pages = {215--249},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {57M99 (20F34)},
      mrnumber = {0786348},
      mrreviewer = {K. Vogtmann},
      doi = {10.2307/1971172},
      zblnumber = {0579.57005},
     }
  • [HatcherWahl] Go to document A. Hatcher and N. Wahl, "Stabilization for mapping class groups of 3-manifolds," Duke Math. J., vol. 155, iss. 2, pp. 205-269, 2010.
    @article{HatcherWahl, mrkey = {2736166},
      author = {Hatcher, Allen and Wahl, Nathalie},
      title = {Stabilization for mapping class groups of 3-manifolds},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {155},
      year = {2010},
      number = {2},
      pages = {205--269},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {57M07 (20F28)},
      mrnumber = {2736166},
      mrreviewer = {Mihalis A. Sykiotis},
      doi = {10.1215/00127094-2010-055},
      zblnumber = {1223.57004},
     }
  • [hurwitz] A. Hurwitz, "Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten," Math. Ann., vol. 39, pp. 1-60, 1891.
    @article{hurwitz, title = {{Ü}ber {R}iemann'sche {F}l{ä}chen mit gegebenen {V}erzweigungspunkten},
      author = {Hurwitz, Adolf},
      journal = {Math. Ann.},
      volume = {39},
      pages = {1--60},
      year = {1891},
      publisher = {Springer},
      address = {Berlin},
      jfmnumber = {23.0429.01},
      }
  • [KL] N. M. Katz and S. Lang, "Finiteness theorems in geometric classfield theory," Enseign. Math., vol. 27, iss. 3-4, pp. 285-319 (1982), 1981.
    @article{KL, mrkey = {0659153},
      author = {Katz, Nicholas M. and Lang, Serge},
      title = {Finiteness theorems in geometric classfield theory},
      note = {with an appendix by Kenneth A. Ribet},
      journal = {Enseign. Math.},
      fjournal = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
      volume = {27},
      year = {1981},
      number = {3-4},
      pages = {285--319 (1982)},
      issn = {0013-8584},
      coden = {ENMAAR},
      mrclass = {14E20 (12A90 14K15)},
      mrnumber = {0659153},
      mrreviewer = {Ernst-Wilhelm Zink},
      zblnumber = {0495.14011},
      }
  • [knud:projectivityII] Go to document F. F. Knudsen, "The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$," Math. Scand., vol. 52, iss. 2, pp. 161-199, 1983.
    @article{knud:projectivityII, mrkey = {0702953},
      author = {Knudsen, Finn F.},
      title = {The projectivity of the moduli space of stable curves. {II}. {T}he stacks {$M\sb{g,n}$}},
      journal = {Math. Scand.},
      fjournal = {Mathematica Scandinavica},
      volume = {52},
      year = {1983},
      number = {2},
      pages = {161--199},
      issn = {0025-5521},
      coden = {MTSCAN},
      mrclass = {14H10 (14D20 14D22)},
      mrnumber = {0702953},
      mrreviewer = {P. E. Newstead},
      zblnumber = {0544.14020},
      url = {http://www.mscand.dk/article/view/12001/10017},
      }
  • [mall:clroots] Go to document G. Malle, "Cohen-Lenstra heuristic and roots of unity," J. Number Theory, vol. 128, iss. 10, pp. 2823-2835, 2008.
    @article{mall:clroots, mrkey = {2441080},
      author = {Malle, Gunter},
      title = {Cohen-{L}enstra heuristic and roots of unity},
      journal = {J. Number Theory},
      fjournal = {Journal of Number Theory},
      volume = {128},
      year = {2008},
      number = {10},
      pages = {2823--2835},
      issn = {0022-314X},
      coden = {JNUTA9},
      mrclass = {11R29 (11R16)},
      mrnumber = {2441080},
      mrreviewer = {Michael J. Jacobson, Jr.},
      doi = {10.1016/j.jnt.2008.01.002},
      zblnumber = {1225.11143},
     }
  • [Matsumura] H. Matsumura, Commutative Ring Theory, Cambridge: Cambridge Univ. Press, 1986, vol. 8.
    @book{Matsumura, mrkey = {0879273},
      author = {Matsumura, Hideyuki},
      title = {Commutative {R}ing {T}heory},
      series = {Cambridge Stud. Adv. Math.},
      volume = {8},
      note = {translated from the Japanese by M.~Reid},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
      year = {1986},
      pages = {xiv+320},
      isbn = {0-521-25916-9},
      mrclass = {13-01},
      mrnumber = {0879273},
      mrreviewer = {W. V. Vasconcelos},
      zblnumber = {0603.13001},
      }
  • [MH] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, New York: Springer-Verlag, 1973, vol. 73.
    @book{MH, mrkey = {0506372},
      author = {Milnor, John and Husemoller, Dale},
      title = {Symmetric Bilinear Forms},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {73},
      publisher = {Springer-Verlag},
      year = {1973},
      pages = {viii+147},
      mrclass = {15A63 (10C05 57D65)},
      mrnumber = {0506372},
      mrreviewer = {Louis H. Kauffman},
      address = {New York},
      zblnumber = {0292.10016},
      }
  • [palais] Go to document R. S. Palais, "Local triviality of the restriction map for embeddings," Comment. Math. Helv., vol. 34, pp. 305-312, 1960.
    @article{palais, mrkey = {0123338},
      author = {Palais, Richard S.},
      title = {Local triviality of the restriction map for embeddings},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {34},
      year = {1960},
      pages = {305--312},
      issn = {0010-2571},
      mrclass = {57.20},
      mrnumber = {0123338},
      mrreviewer = {M. W. Hirsch},
      URL = {http://retro.seals.ch/digbib/view?pid=com-001:1960:34::24},
      zblnumber = {0207.22501},
      }
  • [Quillen] D. Quillen, "Finite generation of the groups $K_{i}$ of rings of algebraic integers," in Algebraic $K$-theory, I: Higher $K$-theories, New York: Springer-Verlag, 1973, vol. 341, pp. 179-198.
    @incollection{Quillen, mrkey = {0349812},
      author = {Quillen, Daniel},
      title = {Finite generation of the groups {$K\sb{i}$} of rings of algebraic integers},
      booktitle = {Algebraic {$K$}-theory, {I}: {H}igher {$K$}-theories},
      venue = {{P}roc. {C}onf., {B}attelle {M}emorial {I}nst., {S}eattle, {W}ash., 1972},
      pages = {179--198},
      series= {Lecture Notes in Math.},
      volume = {341},
      publisher = {Springer-Verlag},
      year = {1973},
      mrclass = {18F25},
      mrnumber = {0349812},
      mrreviewer = {T. Y. Lam},
      address = {New York},
      zblnumber = {0355.18018},
      }
  • [roma:wewersromagny] M. Romagny and S. Wewers, "Hurwitz spaces," in Groupes de Galois Arithmétiques et Différentiels, Paris: Soc. Math. France, 2006, vol. 13, pp. 313-341.
    @incollection{roma:wewersromagny, mrkey = {2316356},
      author = {Romagny, Matthieu and Wewers, Stefan},
      title = {Hurwitz spaces},
      booktitle = {Groupes de {G}alois Arithmétiques et Différentiels},
      series = {Sémin. Congr.},
      volume = {13},
      pages = {313--341},
      publisher = {Soc. Math. France},
      address = {Paris},
      year = {2006},
      mrclass = {14H30 (14D22)},
      mrnumber = {2316356},
      mrreviewer = {Francesca Vetro},
      zblnumber = {1156.14314},
      }
  • [Rosen] Go to document M. Rosen, "S-units and S-class group in algebraic function fields," J. Algebra, vol. 26, pp. 98-108, 1973.
    @article{Rosen, mrkey = {0327777},
      author = {Rosen, Michael},
      title = {{S}-units and {S}-class group in algebraic function fields},
      journal = {J. Algebra},
      fjournal = {Journal of Algebra},
      volume = {26},
      year = {1973},
      pages = {98--108},
      issn = {0021-8693},
      mrclass = {14H05 (12A45 12A90 14H40)},
      mrnumber = {0327777},
      mrreviewer = {W.-D. Geyer},
      doi = {10.1016/0021-8693(73)90036-7},
      zblnumber = {0265.12003},
      }
  • [Salvetti] Go to document M. Salvetti, "Topology of the complement of real hyperplanes in ${\bf C}^N$," Invent. Math., vol. 88, iss. 3, pp. 603-618, 1987.
    @article{Salvetti, mrkey = {0884802},
      author = {Salvetti, M.},
      title = {Topology of the complement of real hyperplanes in {${\bf C}\sp N$}},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {88},
      year = {1987},
      number = {3},
      pages = {603--618},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {32C40 (55Q52)},
      mrnumber = {0884802},
      mrreviewer = {Richard Randell},
      doi = {10.1007/BF01391833},
      zblnumber = {0594.57009},
      }
  • [Segal] Go to document G. Segal, "The topology of spaces of rational functions," Acta Math., vol. 143, iss. 1-2, pp. 39-72, 1979.
    @article{Segal, mrkey = {0533892},
      author = {Segal, Graeme},
      title = {The topology of spaces of rational functions},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {143},
      year = {1979},
      number = {1-2},
      pages = {39--72},
      issn = {0001-5962},
      coden = {ACMAA8},
      mrclass = {55P10 (32C42 81E10)},
      mrnumber = {0533892},
      mrreviewer = {D. B. Fuks},
      doi = {10.1007/BF02392088},
      zblnumber = {0427.55006},
      }
  • [Wahl] N. Wahl, "Homological stability for mapping class groups of surfaces," in Handbook of Moduli. Vol. III, Int. Press, 2013, vol. 26, pp. 547-583.
    @incollection{Wahl, mrkey = {3135444},
      author = {Wahl, Nathalie},
      title = {Homological stability for mapping class groups of surfaces},
      booktitle = {Handbook of Moduli. {V}ol. {III}},
      series = {Adv. Lect. Math. (ALM)},
      volume = {26},
      pages = {547--583},
      publisher = {Int. Press},
      venue = {Somerville, MA},
      year = {2013},
      mrclass = {57M50 (57R50)},
      mrnumber = {3135444},
      mrreviewer = {Kashyap Rajeevsarathy},
      zblnumber = {1322.57016},
      }
  • [Washington] Go to document L. C. Washington, "Some remarks on Cohen-Lenstra heuristics," Math. Comp., vol. 47, iss. 176, pp. 741-747, 1986.
    @article{Washington, mrkey = {0856717},
      author = {Washington, Lawrence C.},
      title = {Some remarks on {C}ohen-{L}enstra heuristics},
      journal = {Math. Comp.},
      fjournal = {Mathematics of Computation},
      volume = {47},
      year = {1986},
      number = {176},
      pages = {741--747},
      issn = {0025-5718},
      coden = {MCMPAF},
      mrclass = {11R11 (11Y40)},
      mrnumber = {0856717},
      mrreviewer = {K. S. Williams},
      doi = {10.2307/2008187},
      zblnumber = {0627.12002},
      }
  • [wewers] S. Wewers, Construction of Hurwitz spaces, 1998.
    @misc{wewers,
      author = {Stefan Wewers},
      title = {Construction of {H}urwitz spaces},
      note = {{T}hesis, U. Duisburg-Essen},
      year = {1998},
      zblnumber = {0925.14002},
      }
  • [jkyu:monodromy] J. Yu, "Toward a proof of the Cohen-Lenstra conjecture in the function field case," preprint, 1997.
    @article{jkyu:monodromy, title = {Toward a proof of the {C}ohen-{L}enstra conjecture in the function field case},
      author = {Yu, Jiu-Kang},
      journal = {preprint},
      year = {1997},
      }

Authors

Jordan S. Ellenberg

University of Wisconsin, Madison, WI

Akshay Venkatesh

Stanford University, Stanford, CA

Craig Westerland

University of Minnesota, Minneapolis, MN