Formal groups and congruences

Vlasenko, Masha

doi:10.1090/tran/7283

Formal groups and congruences
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by Masha Vlasenko PDF

Trans. Amer. Math. Soc. 371 (2019), 883-902

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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