The Artin–Hasse exponential is a power series introduced by Emil Artin and Helmut Hasse in the context of $p$‑adic analysis and cyclotomic extensions. For a fixed prime number $p$, it is defined by
$$ E_p(x)=\exp!\left(\sum_{n=0}^{\infty}\frac{x^{p^{,n}}}{p^{,n}}\right) =\prod_{n=0}^{\infty}\exp!\left(\frac{x^{p^{,n}}}{p^{,n}}\right) \in \mathbb{Q}[[x]] . $$
Basic properties
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Integral coefficients. Although the exponential function generally introduces denominators, the coefficients of $E_p(x)$ lie in the localization $\mathbb{Z}{(p)}$, i.e. they are $p$‑integral. In particular, $E_p(x)\in \mathbb{Z}{(p)}[[x]]$.
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Radius of convergence. As a formal power series it converges $p$‑adically for all $x$ with $|x|_p\le 1$, and it defines a analytic function on the closed unit disc in the $p$‑adic field $\mathbb{C}_p$.
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Relation to $p$‑adic logarithm. The logarithmic derivative satisfies
$$ \frac{E_p'(x)}{E_p(x)}=\sum_{n=0}^{\infty}x^{p^{,n}-1}, $$
which is a $p$‑adic analogue of the usual logarithmic derivative of $\exp(x)$.
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Multiplicative property. For $x,y$ with $|x|_p,|y|_p\le 1$,
$$ E_p(x+y)=E_p(x)E_p(y)\prod_{k\ge1}E_p!\bigl((x y)^{p^{k-1}}\bigr)^{(-1)^{k-1}} . $$
This identity reflects the fact that $E_p$ interpolates the usual exponential on $p$‑power torsion points of the multiplicative formal group.
Applications
- Construction of $p$‑adic cyclotomic extensions. The series $E_p(\pi)$, where $\pi$ is a uniformizer of a finite extension of $\mathbb{Q}_p$, generates totally ramified abelian extensions, mirroring the role of the complex exponential in classical cyclotomic theory.
- Formal groups. $E_p(x)$ serves as the exponential map of the Lubin–Tate formal group associated with the uniformizer $p$, providing an explicit isomorphism between the additive and multiplicative formal groups over $\mathbb{Z}_p$.
- Iwasawa theory. The coefficients of $E_p(x)$ appear in the study of $p$‑adic $L$‑functions and the construction of norm‑compatible sequences of units (the so‑called “cyclotomic units”).
Historical notes
The series was first written down by Emil Artin in 1934 while investigating the structure of abelian extensions of local fields. Helmut Hasse later refined the construction and proved the integrality of its coefficients, leading to the joint eponym “Artin–Hasse exponential.” Their work laid groundwork for later developments in local class field theory and the theory of formal groups.
References
- Artin, E. (1934). Zur Theorie der $p$‑adischen Zahlen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 10, 27‑63.
- Hasse, H. (1936). Über die Struktur der $p$‑adischen Zahlenkörper. Journal für die reine und angewandte Mathematik, 175, 8‑23.
- Serre, J.-P. (1979). Local Fields. Graduate Texts in Mathematics, vol. 67, Springer.
- Washington, L. C. (1997). Introduction to Cyclotomic Fields, 2nd ed., Springer.
The Artin–Hasse exponential remains a fundamental tool in $p$‑adic number theory, providing a bridge between additive and multiplicative structures in the arithmetic of local fields.
