Algebraic gamma monomials and double coverings of cyclotomic fields

Das, Pinaki

doi:10.1090/S0002-9947-00-02417-X

Algebraic gamma monomials and double coverings of cyclotomic fields
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by Pinaki Das PDF

Trans. Amer. Math. Soc. 352 (2000), 3557-3594 Request permission

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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