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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Algebraic gamma monomials and double coverings of cyclotomic fields


Author: Pinaki Das
Journal: Trans. Amer. Math. Soc. 352 (2000), 3557-3594
MSC (1991): Primary 11R18; Secondary 11R32, 11G99
Published electronically: March 28, 2000
MathSciNet review: 1638625
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Abstract: We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex ${\mathbb{SK} }$, to compute $H^*(\pm, {\mathbb{U} })$, where ${\mathbb{U} }$ is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension $K/F$, we define a double covering of $K/F$ to be an extension $\tilde{K}/K$ of degree $\leq 2$, such that ${\tilde{K}}/F$ is Galois. We demonstrate that each class ${\mathbf{a}}\in H^2(\pm, {\mathbb{U} }) $ gives rise to a double covering of ${\mathbb{Q} }(\zeta_ \infty)/{\mathbb{Q} }$, by ${\mathbb{Q} }(\zeta_ \infty,\sqrt{\sin{\mathbf{a}}})/{\mathbb{Q} }(\zeta_ \infty)$. When ${\mathbf{a}}$ lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if ${\mathbf{a}}$ represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of ${\mathbb{Q} }(\zeta_ \infty,\sqrt{\sin{\mathbf{a}}})/{\mathbb{Q} }$ is abelian and hence $\sqrt{\sin{\mathbf{a}}} \in {\mathbb{Q} }(\zeta_ \infty)$. The $\sqrt{\sin{\mathbf{a}}}$ may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.


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Additional Information

Pinaki Das
Affiliation: Department of Mathematics, Pennsylvania State University, McKeesport, Pennsylvania 15132
Email: pxd14@psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02417-X
PII: S 0002-9947(00)02417-X
Received by editor(s): September 18, 1997
Received by editor(s) in revised form: June 29, 1998
Published electronically: March 28, 2000
Article copyright: © Copyright 2000 American Mathematical Society