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CM Periods, CM Regulators, and Hypergeometric Functions, I

Published online by Cambridge University Press:  20 November 2018

Masanori Asakura  and
Noriyuki Otsubo
Masanori Asakura
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo, 060-0810 Japan email: asakura@math.sci.hokudai.ac.jp
Noriyuki Otsubo
Affiliation:
Department of Mathematics and Informatics, Chiba University, Chiba, 263-8522 Japan email: otsubo@math.s.chiba-u.ac.jp
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Abstract

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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.

Keywords

14D0719F2733C2011G1514K22periodregulatorcomplex multiplicationhypergeometric function
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 3 , 01 June 2018 , pp. 481 - 514
Copyright
Copyright © Canadian Mathematical Society 2018

References

[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] 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] Asakura, M., A formula for Beilinson's regulator map on K1 of a fibration of curves having a totally degenerate semistable fiber. arxiv:1310.2810.Google Scholar
[4] Asakura, M. and Otsubo, N., CM periods, CM regulators and hypergeometric functions. II. arxiv:1 503.08894.Google Scholar
[5] Asakura, M. and Sato, K., Chern classes and Riemann-Roch theorem for cohomology without homotopy invariance. arxiv:1301.5829Google Scholar
[6] Barth, W., Hulek, K., Peters, C., and Van de Ven, A., Compact complex surfaces. Second, ed. Springer-Verlag, Berlin, 2004.Google Scholar
[7] Beilinson, A. A., Higher regulators and values of L-functions. J. Soviet Math. 30(1985), 2036–2070.Google Scholar
[8] Chowla, S. and Selberg, A., On Epstein's zeta-function. J. Reine Angew. Math. 227(1967), 86–110.Google Scholar
[9] Deligne, P., Equations différentielles à points singuliers réguliers, Lect. Notes Math. 163, Springer, 1970.Google Scholar
[10] Deligne, P., Théorie de Hodge. III. Publ. Math. Inst. Hautes Études Sci. Publ. Math. 44(1974), 5–77.Google Scholar
[11] Erdélyi, A. et al. ed., Higher transcendental functions. Vol. 1. California Inst. Tech, 1981.Google Scholar
[12] Fresán, J., Periods of Hodge structures and special values of the gamma function. Invent. Math. 208(2017), 247–282. http://dx.doi.org/10.1007/s00222-016-0690-4 Google Scholar
[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. http://dx.doi.org/10.1007/BF01390273 Google Scholar
[14] Hartshorne, R., On the de Rham cohomology of algebraic varieties. Publ. Math. IHES 45(1975), 5–99.Google Scholar
[15] Lerch, M., Sur quelques formules relatives au nombre des classes. Bull. Sci. Math. 21(1897), prem. partie, 290–304.Google Scholar
[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] Morrison, D. R., The Clemens-Schmid exact sequence and applications. In: Topics in transcendental algebraic geometry, Griffiths, P., ed. Ann. Math. Studies, 106. Princeton Univ. Press, Princeton NJ, 1984, pp. 101–119.Google Scholar
[18] Otsubo, N., On the regulator of Fermat motives and generalized hypergeometric functions. J. Reine Angew. Math. 660(2011), 27–82.Google Scholar
[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] 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] Saito, T., Vanishing cycles and geometry of curves over a discrete valuation ring. Amer. J. Math. 109(1987), no. 6, 1043–1085. http://dx.doi.org/10.2307/2374585 Google Scholar
[22] Saito, T. and Terasoma, T., Determinant of period integrals. J. Amer. Math. Soc. 10(1997), 865–937. http://dx.doi.org/10.1090/S0894-0347-97-00243-9 Google Scholar
[23] Shimura, G., Automorphic forms and periods of abelian varieties. J. Math. Soc. Japan 31(1979), 561–592. http://dx.doi.org/10.2969/jmsj703130561 Google Scholar
[24] Slater, L. J., Generalized hypergeometric functions. Cambridge University Press, Cambridge, 1966.Google Scholar
[25] Steenbrink, J., Limits of Hodge structures. Invent. Math. 31(1976), 229–257. http://dx.doi.Org/10.1007/BF01403146 Google Scholar
[26] Steenbrink, J. and Zucker, S., Variation of mixed Hodge structure. I. Invent. Math. 80(1985), 489–542. http://dx.doi.org/10.1007/BF01388729 Google Scholar