No CrossRef data available.
Article contents
Homology of the Fermat Tower and Universal Measures for Jacobi Sums
Published online by Cambridge University Press: 20 November 2018
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
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.
- Type
- Research Article
- Information
- 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
You have
Access