Geodesic
CM Periods, CM Regulators, and Hypergeometric Functions, I
Canadian journal of mathematics, Tome 70 (2018) no. 3, pp. 481-514

Voir la notice de l'article provenant de la source Cambridge University Press

We prove the Gross–Deligne conjecture on CM periods for motives associated with ${{H}^{2}}$ of certain surfaces fibered over the projective line. Then we prove for the same motives a formula which expresses the ${{K}_{1}}$ -regulators in terms of hypergeometric functions $_{3}{{F}_{2}}$ , and obtain a new example of non-trivial regulators.
DOI : 10.4153/CJM-2017-008-6
Mots-clés : 14D07, 19F27, 33C20, 11G15, 14K22, period, regulator, complex multiplication, hypergeometric function 
Asakura, Masanori; Otsubo, Noriyuki. CM Periods, CM Regulators, and Hypergeometric Functions, I. Canadian journal of mathematics, Tome 70 (2018) no. 3, pp. 481-514. doi: 10.4153/CJM-2017-008-6
@article{10_4153_CJM_2017_008_6,
     author = {Asakura, Masanori and Otsubo, Noriyuki},
     title = {CM {Periods,} {CM} {Regulators,} and {Hypergeometric} {Functions,} {I}},
     journal = {Canadian journal of mathematics},
     pages = {481--514},
     year = {2018},
     volume = {70},
     number = {3},
     doi = {10.4153/CJM-2017-008-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-008-6/}
}
TY  - JOUR
AU  - Asakura, Masanori
AU  - Otsubo, Noriyuki
TI  - CM Periods, CM Regulators, and Hypergeometric Functions, I
JO  - Canadian journal of mathematics
PY  - 2018
SP  - 481
EP  - 514
VL  - 70
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-008-6/
DO  - 10.4153/CJM-2017-008-6
ID  - 10_4153_CJM_2017_008_6
ER  - 
%0 Journal Article
%A Asakura, Masanori
%A Otsubo, Noriyuki
%T CM Periods, CM Regulators, and Hypergeometric Functions, I
%J Canadian journal of mathematics
%D 2018
%P 481-514
%V 70
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-008-6/
%R 10.4153/CJM-2017-008-6
%F 10_4153_CJM_2017_008_6

[1] [1] Anderson, G. W., Logarithmic derivatives of Dirichlet L-functions and the periods of Abelian varieties. Compositio Math. 45, Fasc. 3 (1982), 315–332. Google Scholar

[2] [2] Archinard, N., Hypergeometric abelian varieties. Canad. J. Math. 55(2003), 897–932. http://dx.doi.Org/10.4153/CJM-2003-037-4 Google Scholar

[3] [3] Asakura, M., A formula for Beilinson's regulator map on K of a fibration of curves having a totally degenerate semistable fiber. arxiv:1310.2810. Google Scholar

[4] [4] Asakura, M. and Otsubo, N., CM periods, CM regulators and hypergeometric functions. II. arxiv:1 503.08894. Google Scholar

[5] [5] Asakura, M. and Sato, K., Chern classes and Riemann-Roch theorem for cohomology without homotopy invariance. arxiv:1301.5829 Google Scholar

[6] [6] Barth, W., Hulek, K., Peters, C., and Van de Ven, A., Compact complex surfaces. Second, ed. Springer-Verlag, Berlin, 2004. Google Scholar

[7] [7] Beilinson, A. A., Higher regulators and values of L-functions. J. Soviet Math. 30(1985), 2036–2070. Google Scholar

[8] [8] Chowla, S. and Selberg, A., On Epstein's zeta-function. J. Reine Angew. Math. 227(1967), 86–110. Google Scholar

[9] [9] Deligne, P., Equations différentielles à points singuliers réguliers, Lect. Notes Math. 163, Springer, 1970. Google Scholar

[10] [10] Deligne, P., Théorie de Hodge. III. Publ. Math. Inst. Hautes Études Sci. Publ. Math. 44(1974), 5–77. Google Scholar

[11] [11] Erdélyi, A. et al. ed., Higher transcendental functions. Vol. 1. California Inst. Tech, 1981. Google Scholar

[12] [12] Fresán, J., Periods of Hodge structures and special values of the gamma function. Invent. Math. 208(2017), 247–282. Google Scholar | DOI

[13] [13] Gross, B. H. (with an appendix by D. E. Rohrlich), On the periods of Abelian integrals and a formula of Chowla-Selberg. Invent. Math. 45(1978), 193–211. Google Scholar | DOI

[14] [14] Hartshorne, R., On the de Rham cohomology of algebraic varieties. Publ. Math. IHES 45(1975), 5–99. Google Scholar

[15] [15] Lerch, M., Sur quelques formules relatives au nombre des classes. Bull. Sci. Math. 21(1897), prem. partie, 290–304. Google Scholar

[16] [16] Maillot, V. and Roessler, D., On the periods of motives with complex multiplication and a conjecture ofGross-Deligne. Ann. Math. 160(2004), 727–754. http://dx.doi.Org/10.4007/annals.2004.160.727 Google Scholar

[17] [17] Morrison, D. R., The Clemens-Schmid exact sequence and applications. , Griffiths, P., ed. Ann. Math. Studies, 106. Princeton Univ. Press, Princeton NJ, 1984, pp. 101–119. Google Scholar

[18] [18] Otsubo, N., On the regulator of Fermat motives and generalized hypergeometric functions. J. Reine Angew. Math. 660(2011), 27–82. Google Scholar

[19] [19] Otsubo, N., Certain values of Hecke L-functions and generalized hypergeometric functions. J. Number Theory 131(2011), 648–660. http://dx.doi.Org/10.1016/j.jnt.2O10.10.002 Google Scholar

[20] [20] Otsubo, N., On special values of jacobi-sum Hecke L-functions. Exper. Math. 24(2015), no. 2, 247–259. http://dx.doi.Org/10.1080/10586458.2014.971199 Google Scholar

[21] [21] Saito, T., Vanishing cycles and geometry of curves over a discrete valuation ring. Amer. J. Math. 109(1987), no. 6, 1043–1085. Google Scholar | DOI

[22] [22] Saito, T. and Terasoma, T., Determinant of period integrals. J. Amer. Math. Soc. 10(1997), 865–937. Google Scholar | DOI

[23] [23] Shimura, G., Automorphic forms and periods of abelian varieties. J. Math. Soc. Japan 31(1979), 561–592. Google Scholar | DOI

[24] [24] Slater, L. J., Generalized hypergeometric functions. Cambridge University Press, Cambridge, 1966. Google Scholar

[25] [25] Steenbrink, J., Limits of Hodge structures. Invent. Math. 31(1976), 229–257. http://dx.doi.Org/10.1007/BF01403146 Google Scholar

[26] [26] Steenbrink, J. and Zucker, S., Variation of mixed Hodge structure. I. Invent. Math. 80(1985), 489–542. Google Scholar | DOI

Cité par Sources :