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The relative Lubin-Tate formal groups and $$p$$-adic interpolating functions. (English) Zbl 0909.11053
The author extends the results of Shiratani and Imada, who defined the number $$B_n(F,h)$$ and constructed a $$p$$-adic interpolating function to explain many $$p$$-adic interpolating functions in a unified manner [K. Shiratani and T. Imada, Mem. Fac. Sci., Kyushu Univ., Ser. A 46, 351-365 (1992; Zbl 0777.11050)]. He now considers the relative formal group $$F$$ over the ring of integers of a finite unramified extension of $${\mathbb Q}_p$$ (the $$p$$-adic field) instead of the formal group over $$p$$-adic integers. For any primitive Dirichlet character $$\xi$$ with conductor a power of $$p$$, he defines the numbers $$B_{n, \xi}(F, h)$$ as analogues of generalized Bernoulli numbers and constructs a $$p$$-adic interpolating function $$L_p(s, \xi , F, h)$$ extending Shiratani’s $$\zeta _p (s, F, h)$$, and finally calculates the value of it and deduces a generalization of Leopoldt’s formula on the value at $$s=1$$ of the $$p$$-adic $$L$$-function.

##### MSC:
 11S31 Class field theory; $$p$$-adic formal groups 14L05 Formal groups, $$p$$-divisible groups 11S40 Zeta functions and $$L$$-functions
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