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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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