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The exponential series of the Lubin-Tate groups and \(p\)-adic interpolation. (English) Zbl 0777.11050
For a Lubin-Tate formal group \(F(X,Y)\) over \(p\)-adic integers with the logarithm \(\lambda_ F\) and exponential \(e_ F\), and an invertible series \(h(X)\in {\mathcal O}((X))^*\), where \({\mathcal O}\) is the ring of integers in the completion of the algebraic closure of the \(p\)-adic field, the authors introduce generalized Bernoulli numbers \(B_ n(F,h)\) as \(Xh'(e_ F(X))/(\lambda_ F'(e_ F(X))h(e_ F(X))) =\sum_{n=0}^ \infty B_ n(F,h)X^ n /n!\).
It is shown that there exists a locally analytic function connected to the \(B_ n(F,h)\) in the same way as the usual \(p\)-adic zeta function arises from Bernoulli numbers. The proof involves certain more or less elementary calculations of the coefficients of \(e_ F,\lambda_ F\).
Related considerations of \(e_ F,\lambda_ F\) can be found in S. V. Vostokov [Vestn. Leningr. Univ., Ser. I 1988, No. 1, 14-17 (1988); English translation in Vestn. Leningr. Univ., Math. 21, No. 1, 16-20 (1988; Zbl 0649.12013); Izv. Akad. Nauk SSSR, Ser. Mat. 45, No. 5, 985- 1014 (1981); English translation in Math. USSR, Izv. 19, 261-284 (1982; Zbl 0483.14014), section 11] and C. Snyder [Rocky Mount. J. Math. 23, 339-351 (1993)].
Reviewer: I.Fesenko (Bonn)

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11S31 Class field theory; \(p\)-adic formal groups
14L05 Formal groups, \(p\)-divisible groups
11S40 Zeta functions and \(L\)-functions
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