Abstract
Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k’$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre $X_K$ and special fibre $X_k$. Assume that $X_K$ is geometrically connected and of Hodge type $\geq 1$ in positive degrees. Then we show that the number of $k’$-rational points of $X$ satisfies the congruence $|X(k’)| \equiv 1$ mod $|k’|$. We deduce such congruences from a vanishing theorem for the Witt cohomology groups $H^q(X_k, W\mathcal{O}_{X_k,\mathbb{Q}})$ for $q > 0$. In our proof of this last result, a key step is the construction of a trace morphism between the Witt cohomologies of the special fibres of two flat regular $R$-schemes $X$ and $Y$ of the same dimension, defined by a surjective projective morphism $f : Y \to X$.
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ZBLNUMBER = {0258.14006},
DOI = {10.1090/S0002-9904-1972-12976-8},
} -
[Ma73]
B. Mazur, "Frobenius and the Hodge filtration (estimates)," Ann. of Math., vol. 98, pp. 58-95, 1973.
@article {Ma73, MRKEY = {0321932},
AUTHOR = {Mazur, B.},
TITLE = {Frobenius and the {H}odge filtration (estimates)},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {98},
YEAR = {1973},
PAGES = {58--95},
ISSN = {0003-486X},
MRCLASS = {14G20 (14F30 14G05)},
MRNUMBER = {0321932},
MRREVIEWER = {Gerhard Pfister},
ZBLNUMBER = {0261.14005},
DOI = {10.2307/1970906},
} -
[Na09] Y. Nakkajima, Weight filtration and slope filtration on the rigid cohomology of a variety, 2009.
@misc{Na09,
author={Nakkajima, Y.},
TITLE={Weight filtration and slope filtration on the rigid cohomology of a variety},
NOTE={to appear in {\it Mém. Soc. Math. France}},
YEAR={2009},
} -
[Ol07] M. C. Olsson, Crystalline Cohomology of Algebraic Stacks and Hyodo-Kato Cohomology, , 2007, vol. 316.
@book {Ol07, MRKEY = {2451400},
AUTHOR = {Olsson, Martin C.},
TITLE = {Crystalline Cohomology of Algebraic Stacks and {H}yodo-{K}ato Cohomology},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {316},
YEAR = {2007},
PAGES = {412 pp. (2008)},
ISSN = {0303-1179},
ISBN = {978-2-85629-249-5},
MRCLASS = {14F30 (11G25 14A20 14F20 14G20)},
MRNUMBER = {2451400},
MRREVIEWER = {Michel Gros},
ZBLNUMBER = {1199.14006},
} -
[Ra70]
M. Raynaud, "Spécialisation du foncteur de Picard," Inst. Hautes Études Sci. Publ. Math., vol. 38, pp. 27-76, 1970.
@article {Ra70, MRKEY = {0282993},
AUTHOR = {Raynaud, M.},
TITLE = {Spécialisation du foncteur de {P}icard},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {38},
YEAR = {1970},
PAGES = {27--76},
ISSN = {0073-8301},
MRCLASS = {14.55},
MRNUMBER = {0282993},
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ZBLNUMBER = {0207.51602},
DOI = {10.1007/BF02684651},
} -
[Se58] J. Serre, "Sur la topologie des variétés algébriques en caractéristique $p$," in Symposium Internacional de Topolog\’\i a Algebraica [International Symposium on Algebraic Topology], Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24-53.
@incollection {Se58, MRKEY = {0098097},
AUTHOR = {Serre, Jean-Pierre},
TITLE = {Sur la topologie des variétés algébriques en caractéristique {$p$}},
BOOKTITLE = {Symposium Internacional de Topolog\'\i a Algebraica [{I}nternational Symposium on Algebraic Topology]},
PAGES = {24--53},
PUBLISHER = {Universidad Nacional Autónoma de México and UNESCO, Mexico City},
YEAR = {1958},
MRCLASS = {14.00},
MRNUMBER = {0098097},
MRREVIEWER = {M. Rosenlicht},
ZBLNUMBER = {0098.13103},
} -
@book {SGA1, MRKEY = {0354651},
KEY={SGA1},
TITLE = {Revêtements Étales et Groupe Fondamental},
NOTE = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois Marie 1960--1961 (SGA 1), Dirig{é} par Alexandre Grothendieck. Augment{é} de deux expos{é}s de M. Raynaud},
SERIES={Lecture Notes in Math.},
VOLUME={224},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1971},
PAGES = {xxii+447},
MRCLASS = {14-06 (14E20)},
MRNUMBER = {0354651},
DOI = {10.1007/BFb0058656},
} -
[SGA4II]
Théorie des Topos et Cohomologie Étale des Schémas. Tome 2, New York: Springer-Verlag, 1972, vol. 270.
@book {SGA4II, KEY={SGA4II},
MRKEY = {0354653},
TITLE = {Théorie des Topos et Cohomologie Étale des Schémas. {T}ome 2},
SERIES = {Lecture Notes in Math.},
VOLUME={270},
NOTE = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie 1963--1964 (SGA 4), Dirig{é} par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1972},
PAGES = {iv+418},
MRCLASS = {14-06},
MRNUMBER = {0354653},
DOI = {10.1007/BFb0061319},
ZBLNUMBER = {0237.00012},
} -
@book {SGA412, MRKEY = {0463174},
KEY={SGA4$\frac12$},
AUTHOR={Deligne, P.},
TITLE = {Cohomologie Étale},
SERIES = {Lecture Notes in Math.},
VOLUME={569},
NOTE = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie SGA 4${\frac{1}{2}}$, Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1977},
PAGES = {iv+312pp},
MRCLASS = {14F20},
MRNUMBER = {0463174},
MRREVIEWER = {J. S. Milne},
ZBLNUMBER = {0349.14008},
ZBLNUMBER = {0345.00010},
DOI = {10.1007/BFb0091516}
} -
[SGA6]
Théorie des Intersections et Théorème de Riemann-Roch, New York: Springer-Verlag, 1971, vol. 225.
@book {SGA6, MRKEY = {0354655},
KEY={SGA6},
TITLE = {Théorie des Intersections et Théorème de {R}iemann-{R}och},
SERIES = {Lecture Notes in Math.},
VOLUME={225},
NOTE = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie 1966--1967 (SGA 6), Dirig{é} par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1971},
PAGES = {xii+700},
MRCLASS = {14-06},
MRNUMBER = {0354655},
DOI = {10.1007/BFb0066283},
ZBLNUMBER = {0218.14001},
} -
[SGA7II]
Groupes de Monodromie en Géométrie Algébrique. II, New York: Springer-Verlag, 1973, vol. 340.
@book {SGA7II, MRKEY = {0354657},
KEY={SGA7II},
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), Dirig{é} par P. Deligne et N. Katz},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1973},
PAGES = {x+438},
MRCLASS = {14-06},
MRNUMBER = {0354657},
DOI = {10.1007/BFb0060505},
ZBLNUMBER = {0258.00005},
} -
[Ts98] T. Tsuji, "$p$-adic Hodge theory in the semi-stable reduction case," in Proceedings of the International Congress of Mathematicians, Vol. II, 1998, pp. 207-216.
@inproceedings {Ts98, MRKEY = {1648071},
AUTHOR = {Tsuji, Takeshi},
TITLE = {{$p$}-adic {H}odge theory in the semi-stable reduction case},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {II}},
VENUE={{B}erlin, 1998},
SERIES = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
YEAR = {1998},
VOLUME = {extra volume II},
PAGES = {207--216},
ISSN = {1431-0635},
MRCLASS = {14F30},
MRNUMBER = {1648071},
MRREVIEWER = {Guy Laffaille},
ZBLNUMBER = {0937.14011},
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[Ts99]
T. Tsuji, "$p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case," Invent. Math., vol. 137, iss. 2, pp. 233-411, 1999.
@article {Ts99, MRKEY = {1705837},
AUTHOR = {Tsuji, Takeshi},
TITLE = {{$p$}-adic étale cohomology and crystalline cohomology in the semi-stable reduction case},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {137},
YEAR = {1999},
NUMBER = {2},
PAGES = {233--411},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14F30 (14F20)},
MRNUMBER = {1705837},
MRREVIEWER = {Abdellah Mokrane},
DOI = {10.1007/s002220050330},
ZBLNUMBER = {0945.14008},
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[Tz04]
N. Tsuzuki, "Cohomological descent in rigid cohomology," in Geometric Aspects of Dwork Theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 931-981.
@incollection {Tz04, MRKEY = {2099093},
AUTHOR = {Tsuzuki, Nobuo},
TITLE = {Cohomological descent in rigid cohomology},
BOOKTITLE = {Geometric Aspects of {D}work Theory. {V}ol. {I},
{II}},
PAGES = {931--981},
PUBLISHER = {Walter de Gruyter GmbH \& Co. KG, Berlin},
YEAR = {2004},
MRCLASS = {14F30 (14F20 14F40)},
MRNUMBER = {2099093},
MRREVIEWER = {Elmar Grosse-Kl{ö}nne},
ZBLNUMBER = {1073.14026},
DOI = {10.1515/9783110198133},
}