Abstract
Nous partons d’une série $F=\sum_{r\gg -\infty} f_r \cdot\varpi^r$ où $\varpi$ est l’indéterminée et les coefficients $f_r=f_r(z_1,\dots, z_n)$ sont des fonctions holomorphes définies sur un voisinage ouvert du polydisque fermé $\bar{\mathbb{D}}^n=\{(z_1,\dots,z_n);\, |z_i|\leq 1\}$. En intégrant les coefficients de cette série sur le $n$-cube réel $[0,1]^n$, on obtient la série de Laurent $\int_{[0,1]^n}F$. Lorsque $F$ est algébrique nous dirons que $\int_{[0,1]^n}F$ est une série de périodes. Dans cet article, nous cherchons à déterminer les séries algébriques $F$ telles que $\int_{[0,1]^n}F$ est nulle. En principle, ceci fournit des informations sur les propriétés de transcendance des séries de périodes. Notre résultat principal rappelle la conjecture des périodes de Kontsevich-Zagier sous une forme remaniée.
We start with a series $F=\sum_{r\gg -\infty} f_r \cdot\varpi^r$ with indeterminate $\varpi$ and where the coefficients $f_r=f_r(z_1,\dots,z_n)$ are holomorphic functions defined on an open neighborhood of the closed polydisc $\bar{\mathbb{D}}^n\!=\!\{(z_1,\dots,z_n);\, |z_i|\!\leq\! 1\}$. Integrating the coefficients of this series on the $n$-dimensional real cube $[0,1]^n$ yields a Laurent series $\int_{[0,1]^n}F$. When $F$ is algebraic we say that $\int_{[0,1]^n}F$ is a series of periods. In this article, our goal is to determine the algebraic series $F$ such that $\int_{[0,1]^n}F$ is zero. In principle, this gives informations on the transcendence properties of series of periods. Our main result is reminiscent to the Kontsevich-Zagier conjecture on periods in a modified form.
-
[these-doctorat-I] J. Ayoub, Les Six Opérations de Grothendieck et le Formalisme des Cycles Évanescents dans le Monde Motivique. I, Paris: Math. Soc. France, 2007, vol. 314.
@book{these-doctorat-I, mrkey = {2423375},
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title = {Les Six Opérations de {G}rothendieck et le Formalisme des Cycles Évanescents dans le Monde Motivique. {I}},
series = {Astérisque},
fjournal = {Astérisque},
volume = {314},
year = {2007},
pages = {x+466 pp. (2008)},
issn = {0303-1179},
isbn = {978-2-85629-244-0},
mrclass = {14F20 (14C25 14F42 18A40 18F10 18F20 18G55 19E15)},
mrnumber = {2423375},
mrreviewer = {Christian Haesemeyer},
publisher = {Math. Soc. France},
address = {Paris},
zblnumber = {1146.14001},
} -
[these-doctorat-II] J. Ayoub, Les Six Opérations de Grothendieck et le Formalisme des Cycles Évanescents dans le Monde Motivique. II, Paris: Math. Soc. France, 2007, vol. 315.
@book{these-doctorat-II, mrkey = {2438151},
author = {Ayoub, Joseph},
title = {Les Six Opérations de {G}rothendieck et le Formalisme des Cycles Évanescents dans le Monde Motivique. {II}},
series = {Astérisque},
fjournal = {Astérisque},
volume = {315},
year = {2007},
pages = {vi+364 pp. (2008)},
issn = {0303-1179},
isbn = {978-2-85629-245-7},
mrclass = {14C25 (14F20 14F42 18A40 19E15)},
publisher = {Math. Soc. France},
address = {Paris},
mrnumber = {2438151},
mrreviewer = {Christian Haesemeyer},
zblnumber = {1153.14001},
} -
[real-betti]
J. Ayoub, "Note sur les opérations de Grothendieck et la réalisation de Betti," J. Inst. Math. Jussieu, vol. 9, iss. 2, pp. 225-263, 2010.
@article{real-betti, mrkey = {2602027},
author = {Ayoub, Joseph},
title = {Note sur les opérations de {G}rothendieck et la réalisation de {B}etti},
journal = {J. Inst. Math. Jussieu},
fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
volume = {9},
year = {2010},
number = {2},
pages = {225--263},
issn = {1474-7480},
mrclass = {14F42 (14F20 14F25 18F10 18F20 18G55)},
mrnumber = {2602027},
mrreviewer = {Jakob Scholbach},
doi = {10.1017/S1474748009000127},
zblnumber = {1202.14018},
} -
[mot-rig] J. Ayoub, Motifs des variétés analytiques rigides.
@misc{mot-rig,
author = {Ayoub, Joseph},
title = {Motifs des variétés analytiques rigides},
note = {preprint, to appear in {\it Mém. Soc. Math. Fr.} {\bf 141/142} (2015)},
sortyear = {2015},
} -
[gal-mot-1]
J. Ayoub, "L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle, I," J. reine angew. Math., vol. 693, pp. 1-149, 2014.
@article{gal-mot-1,
author = {Ayoub, Joseph},
title = {L'algèbre de {H}opf et le groupe de {G}alois motiviques d'un corps de caractéristique nulle, {I}},
journal = {J. reine angew. Math.},
volume = {693},
pages = {1--149},
year = {2014},
doi = {10.1515/crelle-2012-0089},
mrnumber = {3259031},
SORTYEAR={2013},
zblnumber = {06339932},
} -
[gal-mot-2]
J. Ayoub, "L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle, II," J. Reine angew. Math., vol. 693, pp. 151-226, 2014.
@article{gal-mot-2,
author = {Ayoub, Joseph},
title = {L'algèbre de {H}opf et le groupe de {G}alois motiviques d'un corps de caractéristique nulle, {II}},
JOURNAL = {J. Reine angew. Math.},
FJOURNAL = {Journal für die Reine und angewandte Mathematik},
VOLUME = {693},
YEAR = {2014},
PAGES = {151--226},
DOI = {10.1515/crelle-2012-0090},
mrnumber = {3259032},
SORTYEAR={2013},
zblnumber = {06339934},
} -
[gal-mot-erratum]
J. Ayoub, "Erratum à L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle, II," J. Reine angew. Math., vol. 693, p. e1-e2, 2014.
@article{gal-mot-erratum,
author = {Ayoub, Joseph},
title = {Erratum à {L}'algèbre de {H}opf et le groupe de {G}alois motiviques d'un corps de caractéristique nulle, {II}},
JOURNAL = {J. Reine angew. Math.},
FJOURNAL = {Journal für die Reine und angewandte Mathematik},
VOLUME = {693},
YEAR = {2014},
PAGES = {e1--e2},
DOI = {10.1515/crelle-2013-0018},
mrnumber = {3259035},
SORTYEAR={2014},
zblnumber = {06339933},
} -
[D-module] A. Borel, P. -P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, New York: Academic Press, 1987, vol. 2.
@book{D-module,
author = {A. Borel and P.-P. Grivel and B. Kaup and A. Haefliger and B. Malgrange and F. Ehlers},
title = {Algebraic $D$-modules},
series = {Perspect. Math.},
volume = {2},
publisher = {Academic Press},
address = {New York},
year = {1987},
zblnumber = {0642.32001},
mrnumber = {0882000},
} -
[Local-stable-ho]
D. Cisinski and F. Déglise, "Local and stable homological algebra in Grothendieck abelian categories," Homology, Homotopy Appl., vol. 11, iss. 1, pp. 219-260, 2009.
@article{Local-stable-ho, mrkey = {2529161},
author = {Cisinski, Denis-Charles and D{é}glise, Fr{é}d{é}ric},
title = {Local and stable homological algebra in {G}rothendieck abelian categories},
journal = {Homology, Homotopy Appl.},
fjournal = {Homology, Homotopy and Applications},
volume = {11},
year = {2009},
number = {1},
pages = {219--260},
issn = {1532-0073},
mrclass = {18G55 (14F42 18E15 18G35)},
mrnumber = {2529161},
mrreviewer = {Annette Huber},
url = {http://projecteuclid.org/euclid.hha/1251832567},
zblnumber = {1175.18007},
} -
[singulier-regulier] P. Deligne, Équations Différentielles à Points Singuliers Réguliers, New York: Springer-Verlag, 1970, vol. 163.
@book{singulier-regulier, mrkey = {0417174},
author = {Deligne, Pierre},
title = {\'{E}quations Différentielles à Points Singuliers Réguliers},
series = {Lecture Notes in Math},
volume = {163},
publisher = {Springer-Verlag},
year = {1970},
pages = {iii+133},
mrclass = {14D05 (14C30)},
mrnumber = {0417174},
mrreviewer = {Helmut Hamm},
address = {New York},
zblnumber = {0244.14004},
} -
[cat-tannak-del] P. Deligne, "Catégories tannakiennes," in The Grothendieck Festschrift, Vol. II, Birkhäuser, Boston, 1990, vol. 87, pp. 111-195.
@incollection{cat-tannak-del, mrkey = {1106898},
author = {Deligne, Pierre},
title = {Catégories tannakiennes},
booktitle = {The {G}rothendieck {F}estschrift, {V}ol. {II}},
series = {Progr. Math.},
volume = {87},
pages = {111--195},
publisher = {Birkhäuser, Boston},
year = {1990},
mrclass = {14A99 (12H05 18A99)},
mrnumber = {1106898},
mrreviewer = {James Milne},
zblnumber = {0727.14010},
} -
[FSV] V. Voevodsky, A. Suslin, and E. M. Friedlander, Cycles, Transfers, and Motivic Homology Theories, Princeton, NJ: Princeton Univ. Press, 2000, vol. 143.
@book{FSV, mrkey = {1764197},
author = {Voevodsky, Vladimir and Suslin, Andrei and Friedlander, Eric M.},
title = {Cycles, Transfers, and Motivic Homology Theories},
series = {Ann. of Math. Stud.},
volume = {143},
publisher = {Princeton Univ. Press},
address = {Princeton, NJ},
year = {2000},
pages = {vi+254},
isbn = {0-691-04814-2; 0-691-04815-0},
mrclass = {14F42 (14C25 19E15)},
mrnumber = {1764197},
mrreviewer = {Spencer J. Bloch},
zblnumber = {1021.14006},
} -
[deRham-groth]
A. Grothendieck, "On the De Rham cohomology of algebraic varieties," Inst. Hautes Études Sci. Publ. Math., vol. 29, iss. 29, pp. 95-103, 1966.
@article{deRham-groth, mrkey = {0199194},
author = {Grothendieck, A.},
title = {On the {D}e {R}ham cohomology of algebraic varieties},
journal = {Inst. Hautes Études Sci. Publ. Math.},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
number = {29},
year = {1966},
pages = {95--103},
issn = {0073-8301},
mrclass = {14.55 (32.44)},
mrnumber = {0199194},
mrreviewer = {M. F. Atiyah},
url = {http://www.numdam.org/item?id=PMIHES_1966__29__95_0},
zblnumber = {0145.17602},
volume = {29},
} -
[EGA31]
A. Grothendieck and J. Dieudonné, "Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I," Inst. Hautes Études Sci. Publ. Math., vol. 11, pp. 5-167, 1961.
@article{EGA31,
author = {Grothendieck, A. and Dieudonn{é},
J.},
title = {\'{E}léments de géométrie algébrique. {III}. \'{E}tude cohomologique des faisceaux cohérents. {I}},
journal = {Inst. Hautes Études Sci. Publ. Math.},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
year = {1961},
pages = {5--167},
volume = {11},
mrnumber = {0163910},
zblnumber = {0118.36206},
URL = {http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1961__11__5_0},
} -
[EGA44]
A. Grothendieck and J. Dieudonné, "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie," Inst. Hautes Études Sci. Publ. Math., vol. 32, pp. 1-361, 1967.
@article{EGA44, mrkey = {0238860},
author = {Grothendieck, A. and Dieudonn{é},
J.},
title = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas, {Q}uatrième partie},
journal = {Inst. Hautes Études Sci. Publ. Math.},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {32},
year = {1967},
pages = {1--361},
issn = {0073-8301},
mrclass = {14.55},
mrnumber = {0238860},
mrreviewer = {J. P. Murre},
URL = {http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1967__32__5_0},
zblnumber = {0153.22301},
} -
[kontsevich-periods] M. Kontsevich and D. Zagier, "Periods," in Mathematics Unlimited—2001 and Beyond, New York: Springer-Verlag, 2001, pp. 771-808.
@incollection{kontsevich-periods, mrkey = {1852188},
author = {Kontsevich, Maxim and Zagier, Don},
title = {Periods},
booktitle = {Mathematics Unlimited---2001 and Beyond},
pages = {771--808},
publisher = {Springer-Verlag},
year = {2001},
mrclass = {11-02 (11F67 11G40 11G55)},
mrnumber = {1852188},
mrreviewer = {F. Beukers},
address = {New York},
zblnumber = {1039.11002},
} -
[transfers-deRham]
F. Lecomte and N. Wach, "Le complexe motivique de De Rham," Manuscripta Math., vol. 129, iss. 1, pp. 75-90, 2009.
@article{transfers-deRham, mrkey = {2496957},
author = {Lecomte, Florence and Wach, Nathalie},
title = {Le complexe motivique de {D}e {R}ham},
journal = {Manuscripta Math.},
fjournal = {Manuscripta Mathematica},
volume = {129},
year = {2009},
number = {1},
pages = {75--90},
issn = {0025-2611},
coden = {MSMHB2},
mrclass = {14F40 (14C15)},
mrnumber = {2496957},
mrreviewer = {Feng-Wen An},
doi = {10.1007/s00229-008-0248-x},
zblnumber = {1190.14020},
} -
[survey-mixed]
M. Levine, "Mixed motives," in Handbook of $K$-Theory. Vol. 1, 2, New York: Springer-Verlag, 2005, pp. 429-521.
@incollection{survey-mixed, mrkey = {2181828},
author = {Levine, Marc},
title = {Mixed motives},
booktitle = {Handbook of {$K$}-Theory. {V}ol. 1, 2},
pages = {429--521},
publisher = {Springer-Verlag},
year = {2005},
mrclass = {14F42 (14C15 18E30 19E15)},
mrnumber = {2181828},
mrreviewer = {Annette Huber},
doi = {10.1007/3-540-27855-9_10},
address = {New York},
zblnumber = {1112.14020},
} -
[smooth-mot] M. Levine, "Smooth motives," in Motives and Algebraic Cycles, Amer. Math. Soc., Providence, RI, 2009, vol. 56, pp. 175-231.
@incollection{smooth-mot, mrkey = {2562459},
author = {Levine, Marc},
title = {Smooth motives},
booktitle = {Motives and Algebraic Cycles},
series = {Fields Inst. Commun.},
volume = {56},
pages = {175--231},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2009},
mrclass = {14C15 (14C25 14F42 19E08 19E15 55P42)},
mrnumber = {2562459},
mrreviewer = {Daniel C. Isaksen},
zblnumber = {1183.14016},
} -
[lecture-note-motivic] C. Mazza, V. Voevodsky, and C. Weibel, Lecture Notes on Motivic Cohomology, Providence, RI: Amer. Math. Soc., 2006, vol. 2.
@book{lecture-note-motivic, mrkey = {2242284},
author = {Mazza, Carlo and Voevodsky, Vladimir and Weibel, Charles},
title = {Lecture Notes on Motivic Cohomology},
series = {Clay Math. Monog.},
volume = {2},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2006},
pages = {xiv+216},
isbn = {978-0-8218-3847-1; 0-8218-3847-4},
mrclass = {14F42 (19E15)},
mrnumber = {2242284},
mrreviewer = {Thomas Geisser},
zblnumber = {1115.14010},
} -
[Oliv-Paul-Arne]
O. Röndigs and P. A. Østvaer, "Modules over Motivic cohomology," Adv. Math., vol. 219, iss. 2, pp. 689-727, 2008.
@article{Oliv-Paul-Arne, mrkey = {2435654},
author = {R{ö}ndigs, Oliver and {\O}stv{æ}r, Paul Arne},
title = {Modules over Motivic cohomology},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {219},
year = {2008},
number = {2},
pages = {689--727},
issn = {0001-8708},
coden = {ADMTA4},
mrclass = {14F42 (55U35)},
mrnumber = {2435654},
mrreviewer = {Christian Haesemeyer},
doi = {10.1016/j.aim.2008.05.013},
zblnumber = {1180.14015},
} -
[serre-tate]
. J-P. Serre and J. Tate, "Good reduction of abelian varieties," Ann. of Math., vol. 88, pp. 492-517, 1968.
@article{serre-tate, mrkey = {0236190},
author = {Serre, {\relax J-P} and Tate, John},
title = {Good reduction of abelian varieties},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {88},
year = {1968},
pages = {492--517},
issn = {0003-486X},
mrclass = {14.51},
mrnumber = {0236190},
mrreviewer = {M. J. Greenberg},
doi = {10.2307/1970722},
zblnumber = {0172.46101},
} -
[cancellation-theorem] V. Voevodsky, "Cancellation theorem," Doc. Math., pp. 671-685, 2010.
@article{cancellation-theorem, mrkey = {2804268},
author = {Voevodsky, Vladimir},
title = {Cancellation theorem},
journal = {Doc. Math.},
fjournal = {Documenta Mathematica},
year = {2010},
note = {Extra volume$:$ Andrei A. Suslin sixtieth birthday},
pages = {671--685},
issn = {1431-0635},
mrclass = {14F42 (19E15)},
mrnumber = {2804268},
mrreviewer = {Oliver R{ö}ndigs},
zblnumber = {1202.14022},
} -
[voev-mot-simpl-sch]
V. Voevodsky, "Motives over simplicial schemes," J. K-Theory, vol. 5, iss. 1, pp. 1-38, 2010.
@article{voev-mot-simpl-sch, mrkey = {2600283},
author = {Voevodsky, Vladimir},
title = {Motives over simplicial schemes},
journal = {J. K-Theory},
fjournal = {Journal of K-Theory. K-Theory and its Applications in Algebra, Geometry, Analysis \& Topology},
volume = {5},
year = {2010},
number = {1},
pages = {1--38},
issn = {1865-2433},
mrclass = {14F42 (19E15)},
mrnumber = {2600283},
mrreviewer = {Thomas Geisser},
doi = {10.1017/is010001030jkt107},
zblnumber = {1194.14029},
}