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\(p\)-adic interpolation of the coefficients of Hurwitz series attached to height one formal groups. (English) Zbl 0810.11068
Let \(F\) be a formal group over the ring of integers of a finite extension of \(\mathbb{Q}_ p\), \(\varepsilon(t)\) the formal exponential function of \(F\), and \(f\) a formal power series with coefficients in the ring of integers of \(\mathbb{C}_ p\). Then \(f(\varepsilon (t))= \sum_{k=0}^ \infty c_ k {{t^ k} \over {k!}}\) is called a Hurwitz series attached to \(F\). The author has shown in [Rocky Mt. J. Math. 15, 1-11 (1985; Zbl 0578.14041)] that its coefficients \(c_ k\) satisfy Kummer congruences.
In the present paper he shows that if \(F\) is of height one, certain “twisted” versions \(\widetilde{c}_ k^*\) of the coefficients \(c_ k\) can be \(p\)-adically interpolated by a continuous function \(c(s)\) on \(\mathbb{Z}_ p\) which moreover turns out to be an element of the Iwasawa algebra. Therefore the Kummer congruences for the \(c_ k\) can be deduced from J.-P. Serre’s characterization of the Iwasawa algebra [Modular functions of one variable III, Springer Lect. Notes Math. 350, 191-268 (1973; Zbl 0277.12014)].
MSC:
11S31 Class field theory; \(p\)-adic formal groups
14L05 Formal groups, \(p\)-divisible groups
14G20 Local ground fields in algebraic geometry
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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References:
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