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Homology of the Fermat Tower and Universal Measures for Jacobi Sums

Published online by Cambridge University Press:  20 November 2018

Noriyuki Otsubo
Noriyuki Otsubo*
Affiliation:
Department of Mathematics and Informatics, Chiba University, Chiba, 263-8522 Japan e-mail: [email protected]
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Abstract

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We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a manner similar to Ihara's definition of $\ell$-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

Keywords

11S8011G1511R18Fermat curvesIhara-Anderson theoryJacobi sums
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 624 - 640
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Anderson, G. W., Torsion points on Fermât Jacobians, roots of circular units and relative singular homology. Duke Math. J. 54(1987), no. 2, 501–561. http://dx.doi.org/10.1215/S0012-7094-87-05422-6 Google Scholar
[2] Anderson, G. W., The hyperadelic gamma function: aprécis. In: Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986), Adv. Stud. Pure Math., 12, North-Holland, Amsterdam, 1987, pp. 1–19.Google Scholar
[3] Anderson, G. W., The hyperadelic gamma function. Invent. Math. 95(1989), no. 1, 63–131. http://dx.doi.org/10.1007/BF01394145 Google Scholar
[4] Artin, M., Grothendieck, A., and Verdier, J.-L. (Eds.), Théorie des topos etcohomologie étale des schémas. (SGA 4), Tome 3, Lecture Notes in Mathematics, 305, Springer-Verlag, Berlin-New York, 1973.Google Scholar
[5] Coleman, R. F., Anderson-Iharatheory: Gauss sums and circular units. In: Algebraic number theory, Adv. Stud. Pure Math., 17, Academic Press, Boston, MA, 1989, pp. 55–72.Google Scholar
[6] Griffiths, P. and Harris, J., Principles of algebraic geometry. Pure and applied mathematics. Wiley-Interscience, New York, 1978.Google Scholar
[7] Guàrdia, J., A fundamental domain for the Fermât curves and their quotients. Contributions to the algorithmic study of problems of arithmetic moduli (Spanish).Rev. R. Acad. Cienc. Exactas Fis.Nat. (Esp.) 94(2000), no. 3, 391–396.Google Scholar
[8] Ihara, Y., Profinite braid groups, Galois representations, and complex multiplications. Ann. of Math. 123(1986), no. 1, 43–106. http://dx.doi.org/10.2307/1971352 Google Scholar
[9] Ihara, Y., Braids, Galois groups, and some arithmetic functions. In: Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Mathematical Society of Japan, Tokyo, 1991, pp. 99–120.Google Scholar
[10] Kamata, Y., The algorithm to calculate the period matrix of the curve xm + y” = 1. Tsukuba J. Math. 26(2002), no. 1, 15–37.Google Scholar
[11] Lim, C.-H., Endomorphismsofjacobian varieties of Fermât curves. Compositio Math. 80(1991), no. 1, 85–110.Google Scholar
[12] Otsubo, N., On the regulator of Fermât motives and generalized hypergeometric functions. J. Reine Angew. Math. 660(2011), 27–82.Google Scholar
[13] Otsubo, N., Certain values of Hecke L-functions and generalized hypergeometric functions. J. Number Theory 131(2011), no. 4, 648–660. http://dx.doi.org/10.1016/j.jnt.2010.10.002 Google Scholar
[14] Otsubo, N., On the Abel-Jacobi maps of Fermât Jacobians. Math. Z. 270(2012), 423–444. http://dx.doi.org/10.1007/s00209-010-0805-3 Google Scholar
[15] Rohrlich, D. E., Appendix to: B. H. Gross, On the Periods ofabelian integrals and a formula ofChowla andSelberg. Invent. Math. 45(1978), 193–211. http://dx.doi.org/10.1007/BF01390273 Google Scholar
[16] Weil, A., Jacobi sums as “Grofiencharaktere”. Trans. Amer. Math. Soc. 73(1952), 487–495 Google Scholar
[17] Weil, A., Sur les périodes des intégrales Abéliennes. Comm. Pure App. Math. 29( 1976), no. 6, 813–819. http://dx.doi.org/10.1002/cpa.3160290620 Google Scholar