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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Formal groups and congruences

Author: Masha Vlasenko
Journal: Trans. Amer. Math. Soc. 371 (2019), 883-902
MSC (2010): Primary 14L05; Secondary 11G25, 33C20
Published electronically: July 12, 2018
MathSciNet review: 3885164
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Abstract: We give a criterion of integrality of a one-dimensional formal group law in terms of congruences satisfied by the coefficients of the canonical invariant differential. For an integral formal group law a $ p$-adic analytic formula for the local characteristic polynomial at $ p$ is given. We demonstrate applications of our results to formal group laws attached to $ L$-functions, Artin-Mazur formal groups of algebraic varieties and hypergeometric formal group laws.

This paper was written with the intention to give an explicit and self-contained introduction to the arithmetic of formal group laws, which would be suitable for non-experts. For this reason we consider only one-dimensional laws, though a generalization of our approach to higher dimensions is clearly possible. The ideas of congruences and $ p$-adic continuity in the context of formal groups were considered by many authors. We sketch the relation of our results to the existing literature in a separate paragraph at the end of the introductory section.

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Additional Information

Masha Vlasenko
Affiliation: Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-656, Warsaw

Received by editor(s): September 2, 2016
Received by editor(s) in revised form: March 19, 2017
Published electronically: July 12, 2018
Additional Notes: This work was supported by the National Science Centre of Poland, grant UMO-2016/21/B/ST1/03084.
Article copyright: © Copyright 2018 Masha Vlasenko