Abstract
There should be a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb Z$ with compact support at infinity gives, up to sign, the leading term of the zeta-function of $X$ at $s = 0$. We construct a topology (the Weil-étale topology) for the ring of integers in a number field whose cohomology groups $H^i(\mathbb Z) $ determine such an Euler characterstic if we restrict to $i\leq 3$.
-
[A-T] E. Artin and J. Tate, Class Field Theory, New York: W. A. Benjamin, Inc., 1968.
@book{A-T, MRKEY = {0223335},
AUTHOR = {Artin, E. and Tate, J.},
TITLE = {Class Field Theory},
PUBLISHER = {W. A. Benjamin, Inc.},
ADDRESS = {New York},
YEAR = {1968},
PAGES = {xxvi+259},
MRCLASS = {10.68},
MRNUMBER = {36 \#6383},
ZBLNUMBER = {0176.33504},
} -
[A] M. Artin, "Grothendieck topologies," Harvard University, mimeographed notes , 1962.
@techreport{A,
author = {M. Artin},
TITLE = {Grothendieck topologies},
TYPE = {mimeographed notes},
INSTITUTION = {Harvard University},
YEAR = {1962},
} -
[B-W] A. Borel and N. R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Princeton, N.J.: Princeton Univ. Press, 1980.
@book{B-W, MRKEY = {554917},
AUTHOR = {Borel, Armand and Wallach, Nolan R.},
TITLE = {Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups},
SERIES = {Ann. of Math. Studies},
NUMBER = {94},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1980},
PAGES = {xvii+388},
ISBN = {0-691-08248-0; 0-691-08249-9},
MRCLASS = {22E41 (22E40 22E45 57T15)},
MRNUMBER = {83c:22018},
ZBLNUMBER = {0443.22010},
MRREVIEWER = {Roger Howe},
} -
[D] P. Deligne, "Le déterminant de la cohomologie," in Current Trends in Arithmetical Algebraic Geometry, Providence, RI: Amer. Math. Soc., 1987, pp. 93-177.
@incollection{D, MRKEY = {902592},
AUTHOR = {Deligne, P.},
TITLE = {Le déterminant de la cohomologie},
BOOKTITLE = {Current Trends in Arithmetical Algebraic Geometry},
SERIES = {Contemp. Math.},
NUMBER = {67},
PAGES = {93--177},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1987},
MRCLASS = {32L10 (11G25 14F12 14H99 32G13 58G11)},
MRNUMBER = {89b:32038},
ZBLNUMBER = {0629.14008},
MRREVIEWER = {Daniel S. Freed},
} -
[F]
M. Flach, "Cohomology of topological groups with applications to the Weil group," Compos. Math., vol. 144, iss. 3, pp. 633-656, 2008.
@article{F, MRKEY = {2422342},
AUTHOR = {Flach, M.},
TITLE = {Cohomology of topological groups with applications to the {W}eil group},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {144},
YEAR = {2008},
NUMBER = {3},
PAGES = {633--656},
ISSN = {0010-437X},
MRCLASS = {14Fxx (11G40 18F10 22Axx)},
MRNUMBER = {2422342},
DOI = {10.1112/S0010437X07003338},
ZBLNUMBER = {05292809},
} -
[Ge1]
T. Geisser, "Weil-étale cohomology over finite fields," Math. Ann., vol. 330, iss. 4, pp. 665-692, 2004.
@article{Ge1, MRKEY = {2102307},
AUTHOR = {Geisser, Thomas},
TITLE = {Weil-étale cohomology over finite fields},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {330},
YEAR = {2004},
NUMBER = {4},
PAGES = {665--692},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14F20 (11G25 14F42)},
MRNUMBER = {2006b:14031},
DOI = {10.1007/s00208-004-0564-8},
ZBLNUMBER = {1069.14021},
MRREVIEWER = {Kanetomo Sato},
} -
[Ge2]
T. Geisser, "Arithmetic cohomology over finite fields and special values of $\zeta$-functions," Duke Math. J., vol. 133, iss. 1, pp. 27-57, 2006.
@article{Ge2, MRKEY = {2219269},
AUTHOR = {Geisser, Thomas},
TITLE = {Arithmetic cohomology over finite fields and special values of {$\zeta$}-functions},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {133},
YEAR = {2006},
NUMBER = {1},
PAGES = {27--57},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14F20 (11G25 14F42 14G10)},
MRNUMBER = {2007d:14040},
DOI = {10.1215/S0012-7094-06-13312-4},
ZBLNUMBER = {1104.14011},
MRREVIEWER = {Lei Fu},
} -
[Go] R. Godement, Topologie Algébrique et Théorie des Faisceaux, Paris: Publ. Math. Univ. Strasbourg 13, Hermann, 1958.
@book{Go, MRKEY = {0102797},
AUTHOR = {Godement, Roger},
TITLE = {Topologie Algébrique et Théorie des Faisceaux},
SERIES = {Actualit'es Sci. Ind.},
NUMBER = {1252},
FSERIES = {Actualit'es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13},
PUBLISHER = {{\it Publ. Math. Univ. Strasbourg} {\bf 13},
Hermann},
ADDRESS = {Paris},
YEAR = {1958},
PAGES = {viii+283},
MRCLASS = {55.00},
MRNUMBER = {21 \#1583},
ZBLNUMBER = {0080.16201},
MRREVIEWER = {R. Deheuvels},
} -
[L]
S. Lichtenbaum, "The Weil-étale topology on schemes over finite fields," Compos. Math., vol. 141, iss. 3, pp. 689-702, 2005.
@article{L, MRKEY = {2135283},
AUTHOR = {Lichtenbaum, S.},
TITLE = {The {W}eil-étale topology on schemes over finite fields},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {141},
YEAR = {2005},
NUMBER = {3},
PAGES = {689--702},
ISSN = {0010-437X},
MRCLASS = {14F20 (14G15)},
MRNUMBER = {2006b:14032},
DOI = {10.1112/S0010437X04001150},
ZBLNUMBER = {1073.14024},
MRREVIEWER = {Marco Hien},
} -
[M] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980.
@book{M, MRKEY = {559531},
AUTHOR = {Milne, James 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},
ZBLNUMBER = {0433.14012},
MRREVIEWER = {G. Horrocks},
} -
@article{Mi, MRKEY = {0196736},
AUTHOR = {Milnor, J.},
TITLE = {Whitehead torsion},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical Society},
VOLUME = {72},
YEAR = {1966},
PAGES = {358--426},
ISSN = {0002-9904},
MRCLASS = {55.40 (55.25)},
MRNUMBER = {33 \#4922},
DOI = {10.1090/S0002-9904-1966-11484-2},
ZBLNUMBER = {0147.23104},
MRREVIEWER = {J. F. Adams},
} -
[M1]
C. C. Moore, "Extensions and low dimensional cohomology theory of locally compact groups, I, II," Trans. Amer. Math. Soc., vol. 113, pp. 40-63, 64, 1964.
@article{M1, MRKEY = {0171880},
AUTHOR = {Moore, Calvin C.},
TITLE = {Extensions and low dimensional cohomology theory of locally compact groups, {I},
{II}},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {113},
YEAR = {1964},
PAGES = {40--63, 64--86},
ISSN = {0002-9947},
MRCLASS = {22.60 (28.70)},
MRNUMBER = {30 \#2106},
DOI = {10.2307/1994090},
ZBLNUMBER = {0131.26902},
MRREVIEWER = {J. M. G. Fell},
} -
[M2]
C. C. Moore, "Group extensions and cohomology for locally compact groups: III," Trans. Amer. Math. Soc., vol. 221, iss. 1, pp. 1-33, 1976.
@article{M2, MRKEY = {0414775},
AUTHOR = {Moore, Calvin C.},
TITLE = {Group extensions and cohomology for locally compact groups: {III}},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {221},
YEAR = {1976},
NUMBER = {1},
PAGES = {1--33},
ISSN = {0002-9947},
MRCLASS = {22D05 (22D10 22D30)},
MRNUMBER = {54 \#2867},
DOI = {10.2307/1997540},
ZBLNUMBER = {0366.22005},
MRREVIEWER = {J. M. G. Fell},
} -
[Mo]
P. S. Mostert, "Local cross sections in locally compact groups," Proc. Amer. Math. Soc., vol. 4, pp. 645-649, 1953.
@article{Mo, MRKEY = {0056614},
AUTHOR = {Mostert, Paul S.},
TITLE = {Local cross sections in locally compact groups},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {4},
YEAR = {1953},
PAGES = {645--649},
ISSN = {0002-9939},
MRCLASS = {20.0X},
MRNUMBER = {15,101d},
DOI = {10.2307/2032540},
ZBLNUMBER = {0103.01801},
MRREVIEWER = {A. M. Gleason},
} -
[NSW] J. Neukirch, . A. Schmidt, and . K. Wingberg, Cohomology of Number Fields, New York: Springer-Verlag, 2000.
@book{NSW, MRKEY = {1737196},
AUTHOR = {Neukirch, J. and Schmidt, {\relax\hskip0ptminus1pt A} and Wingberg, {\relax\hskip0ptminus1pt K}},
TITLE = {Cohomology of Number Fields},
SERIES = {Grundl. Math. Wissen.},
NUMBER = {323},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2000},
PAGES = {xvi+699},
ISBN = {3-540-66671-0},
MRCLASS = {11R34 (11-02 11G45 11R23 11S20 11S25 11S31 12G05)},
MRNUMBER = {2000j:11168},
ZBLNUMBER = {0948.11001},
MRREVIEWER = {Gabriel D. Villa-Salvador},
} -
[R]
C. S. Rajan, "On the vanishing of the measurable Schur cohomology groups of Weil groups," Compos. Math., vol. 140, iss. 1, pp. 84-98, 2004.
@article{R, MRKEY = {2004125},
AUTHOR = {Rajan, C. S.},
TITLE = {On the vanishing of the measurable {S}chur cohomology groups of {W}eil groups},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {140},
YEAR = {2004},
NUMBER = {1},
PAGES = {84--98},
ISSN = {0010-437X},
MRCLASS = {11R34 (22E55)},
MRNUMBER = {2004g:11101},
DOI = {10.1112/S0010437X03000046},
ZBLNUMBER = {1057.11051},
MRREVIEWER = {Wee Teck Gan},
} -
[T] J. Tate, "Number theoretic background," in Automorphic Forms, Representations and $L$-Functions, II, Providence, R.I.: Amer. Math. Soc., 1979, pp. 3-26.
@incollection{T, MRKEY = {546607},
AUTHOR = {Tate, J.},
TITLE = {Number theoretic background},
BOOKTITLE = {Automorphic Forms, Representations and {$L$}-Functions, {\rm II}},
VENUE = {Oregon State Univ., Corvallis, OR, 1977},
SERIES = {Proc. Sympos. Pure Math.},
NUMBER = {33},
PAGES = {3--26},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, R.I.},
YEAR = {1979},
MRCLASS = {12A67},
MRNUMBER = {80m:12009},
ZBLNUMBER = {0422.12007},
MRREVIEWER = {A. I. Vinogradov},
} -
[W]
D. Wigner, "Algebraic cohomology of topological groups," Trans. Amer. Math. Soc., vol. 178, pp. 83-93, 1973.
@article{W, MRKEY = {0338132},
AUTHOR = {Wigner, David},
TITLE = {Algebraic cohomology of topological groups},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {178},
YEAR = {1973},
PAGES = {83--93},
ISSN = {0002-9947},
MRCLASS = {18H10 (22A99 57F20)},
MRNUMBER = {49 \#2898},
DOI = {10.2307/1996690},
ZBLNUMBER = {0264.22001},
MRREVIEWER = {A. Kleppner},
}