×

A generalization of the theory of Coleman power series. (English) Zbl 1314.14094

The paper under review is based on the author’s Master‘s Thesis, written under the supervision of S. Kobayashi (Tohoku University, Japan).
The author discusses interesting directions of the generalization of the theory of power series by R. F. Coleman [Invent. Math. 53, 91–116 (1979; Zbl 0429.12010)] in the framework of B. Perrin-Riou [Invent. Math. 115, No. 1, 81–149 (1994; Zbl 0838.11071)] and of S. Kobayashi [Invent. Math. 191, No. 3, 527–629 (2013; Zbl 1300.11053)].
To generalize the theory of Kobayashi [loc. cit.] to general commutative formal groups over unramified rings, one needs the notion of \(Q\)-norm systems. Here, the author uses results by B. Perrin-Riou [Invent. Math. 99, No. 2, 247–292 (1990; Zbl 0715.11030)].
The main result of the paper under review is Theorem 1.1. (cf. Theorem 4.6). This extends results by Coleman, Kobayashi and Perrin-Rion [loc. cit.] to \(d\)-dimensional commutative formal groups over \({\mathbb Z}_p\) of finite height \(h\).
The theorem is proved by a modification of a proof by Kobayashi [loc. cit.] with the help of mention result and the extension of Proposition 2.1 in [H. Knospe, Manuscr. Math. 87, No. 2, 225–258 (1995; Zbl 0847.14026)] to the author’s case.
The paper contains several lemmas and propositions which are of independent interest.

MSC:

14L05 Formal groups, \(p\)-divisible groups
11S31 Class field theory; \(p\)-adic formal groups
11E95 \(p\)-adic theory
13F25 Formal power series rings
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] R. Coleman, Division values in local fields, Invent. Math. 53 (1979), 91-116. · Zbl 0429.12010 · doi:10.1007/BF01390028
[2] J.-M. Fontaine, Groupes \(p\)-divisibles sur les corps locaux, Astérisque, No. 47-48, Société Mathématique de France, Paris, 1977.
[3] T. Honda, On the theory of commutative formal groups, J. Math. Soc. Japan 22 (1970), 213-246. · Zbl 0202.03101 · doi:10.2969/jmsj/02220213
[4] H. Knospe, Iwasawa-theory of abelian varieties at primes of non-ordinary reduction, Manuscripta Math. 87 (1995), 225-258. · Zbl 0847.14026 · doi:10.1007/BF02570472
[5] S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), 1-36. · Zbl 1047.11105 · doi:10.1007/s00222-002-0265-4
[6] S. Kobayashi, The \(p\)-adic Gross-Zagier formula for elliptic curves at supersingular primes, Invent. Math. 191 (2013), 527-629. · Zbl 1300.11053 · doi:10.1007/s00222-012-0400-9
[7] B. Perrin-Riou, Théorie d’Iwasawa \(p\)-adique locale et globale, Invent. Math. 99 (1990), 247-292. · Zbl 0715.11030 · doi:10.1007/BF01234420
[8] B. Perrin-Riou, Théorie d’Iwasawa des représentations \(p\)-adiques sur un corps local, Invent. Math. 115 (1994), 81-149. · Zbl 0838.11071 · doi:10.1007/BF01231755
[9] J. Tate, \(p\)-divisible groups, Proc. Conf. Local Fields, 158-183, Springer, Berlin, 1967. · Zbl 0157.27601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.