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Homology of the Fermat Tower and Universal Measures for Jacobi Sums

Published:2016-04-11
Printed: Sep 2016
• Noriyuki Otsubo,
Department of Mathematics and Informatics, Chiba University, Chiba, 263-8522 Japan
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Abstract

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 similar manner 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: Fermat curves, Ihara-Anderson theory, Jacobi sums
 MSC Classifications: 11S80 - Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 11G15 - Complex multiplication and moduli of abelian varieties [See also 14K22] 11R18 - Cyclotomic extensions

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