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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:
 [1] A. Baker, Lubin-Tate formal groups and $$p$$-adic integration on rings of integers , · JFM 10.0365.04 [2] L. Carlitz, Congruences for the coefficients of the Jacobi elliptic functions , Duke Math. J. 16 (1949), 297-302. · Zbl 0038.17903 [3] T. Honda, Formal groups and zeta functions , Osaka J. Math. 5 (1968), 199-213. · Zbl 0169.37601 [4] A. Hurwitz and R. Courant, Funktionentheorie , J. Springer, Berlin, 1925. [5] N. Katz, $$p$$-adic $$L$$-functions via module of elliptic curves , Algebraic Geometry, Proc. Symposium Pure Math, Arcata, Amer. Math. Soc. 29 , Providence, (1975), 479-506. · Zbl 0317.14009 [6] ——–, Formal groups and $$p$$-adic interpolation , Asterisque 41 -42 (1977), 55-65. · Zbl 0351.14024 [7] ——–, Divisibilities, congruences, and Cartier duality , J. Fac. Sci. Univ., Tokyo, 28 (1982), 667-678. · Zbl 0559.14032 [8] N. Koblitz, $$p$$-adic numbers, $$p$$-adic analysis, and Zeta-functions , Graduate Texts in Math. 58 , Springer-Verlag, New York, 1977. · Zbl 0364.12015 [9] H. Lang, Kummersche Kongruenzen fűr die normierten Entwicklungskoeffizienten der Weierstrassschen-Funktion , Abh. Math. Sem. Univ. 33 , Hamburg, (1969), 183-196. · Zbl 0183.31304 [10] S. Lang, Elliptic functions , Addison Wesley, Reading, MA, 1973. · Zbl 0316.14001 [11] ——–, Cyclotomic fields , Graduate Texts in Math. 59 , Springer-Verlag, New York, 1978. [12] S. Lichlenbaum, On $$p$$-adic $$L$$-functions associated to elliptic curves , Invent. Math. 56 (1980), 19-55. · Zbl 0425.12017 [13] J. Lubin, One parameter formal Lie groups over $$p$$-adic integer rings , Ann. Math. 80 (1964), 464-484. JSTOR: · Zbl 0135.07003 [14] J-P. Serre, Formes modulaires et fonctions zêta $$p$$-adique , Modular functions of one variable III (Antwerp 1972), 191-268, Springer Lecture Notes in Math. 350 (1973). · Zbl 0277.12014 [15] ——–, private communication, 1982. [16] C. Snyder, Kummer congruences for the coefficients of Hurwitz series , Acta Arith. 40 (1982), 175-191. · Zbl 0482.10004 [17] ——–, A concept of Bernoulli numbers in algebraic function fields II, Manuscripta Math. 35 (1981), 69-89. · Zbl 0478.12013 [18] ——–, Kummer congruences in formal groups and algebraic groups of dimension one , Rocky Mountain J. Mathematics 15 , 1985, 1\emdash/11. · Zbl 0578.14041 [19] J. Tate, $$p$$-divisible groups , in Proc. Conf. Local Fields (T.A. Springer, ed.), Springer-Verlag, Berlin (1967), 158-183. · Zbl 0157.27601 [20] L. Washington, Introduction to cyclotomic fields , Graduate Texts in Math. 83 , Springer-Verlag, New York, 1980.
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