The hyperadelic gamma function
The author gives a profinite analogue of the theory of \textit{Y. Ihara} [Ann. Math. (2) 123, 43--106 (1986; Zbl 0595.12003)] concerning the structure of the inverse limit of the \(\ell\)-Tate modules of the Fermat curves of degree \(\ell^ n\). Ihara's methods are not generalized; rather, the author relies on the results of his earlier paper [Duke Math. J. 53, 457--502 (1986; Zbl 0679.14001)]. An adelic beta function is defined which completely describes the action of the Galois group of the rationals on the torsion points of all orders of the Jacobians of the Fermat curves of all degrees. A purely class field theoretic characterization of the adelic beta function is given in which Tate's ``half-transfer'' construction figures prominently. Just as the Eulerian integral of the first kind (the beta function) factors in terms of the Eulerian integral of the second kind (the gamma function), so the adelic beta function factors in terms of the hyperadelic gamma function. The latter is shown to adelically interpolate Gauss sums and to satisfy functional equations analogous to those satisfied by the gamma function itself, e.g. the Gauss multiplication formula. The logarithmic derivative of the hyperadelic gamma function is described in terms of the action of the Galois group of the rationals on the set of roots of all orders of circular units of all orders.
action of the Galois group of the rationals
scientific article; zbMATH DE number 4116688
scientific article