Artin–Hasse exponential
in mathematics, a power series named after Emil Artin and Helmut Hasse
In mathematics, specifically in p-adic analysis, the Artin–Hasse exponential, introduced by Emil Artin and Helmut Hasse in 1928, is the power series given by
E
p
(
x
)
=
exp
(
x
+
x
p
p
+
x
p
2
p
2
+
x
p
3
p
3
+
⋯
)
.
{\displaystyle E_{p}(x)=\exp \left(x+{\frac {x^{p}}{p}}+{\frac {x^{p^{2}}}{p^{2}}}+{\frac {x^{p^{3}}}{p^{3}}}+\cdots \right).}
Motivation
One motivation for considering this series to be analogous to the exponential function comes from infinite products. In the ring of formal power series
Q
[
[
x
]
]
{\displaystyle \mathbb {Q} [[x]]}
we have the identity
e
x
=
∏
n
≥
1
(
1
−
x
n
)
−
μ
(
n
)
/
n
,
{\displaystyle e^{x}=\prod _{n\geq 1}(1-x^{n})^{-\mu (n)/n},}
where
μ
{\displaystyle \mu }
is the Möbius function. This identity can be verified by showing that the logarithmic derivatives of both sides are equal and that both sides have the same constant term. Similarly, one can verify a product expansion for the Artin–Hasse exponential:
E
p
(
x
)
=
∏
(
p
,
n
)
=
1
(
1
−
x
n
)
−
μ
(
n
)
/
n
.
{\displaystyle E_{p}(x)=\prod _{(p,n)=1}(1-x^{n})^{-\mu (n)/n}.}
So passing from a product over all
n
{\displaystyle n}
to a product over only
n
{\displaystyle n}
coprime to
p
{\displaystyle p}
, which is a typical operation in
p
{\displaystyle p}
-adic analysis, leads from
e
x
{\displaystyle e^{x}}
to
E
p
(
x
)
{\displaystyle E_{p}(x)}
.
Properties
The coefficients of
E
p
(
x
)
{\displaystyle E_{p}(x)}
are rational. We can use either formula for
E
p
(
x
)
{\displaystyle E_{p}(x)}
to prove that, unlike
e
x
{\displaystyle e^{x}}
, all of its coefficients are
p
{\displaystyle p}
-integral; in other words, the denominators of the coefficients of
E
p
(
x
)
{\displaystyle E_{p}(x)}
are not divisible by
p
{\displaystyle p}
.
A first proof uses the definition of
E
p
(
x
)
{\displaystyle E_{p}(x)}
and Dwork's lemma, which says that a power series
f
(
x
)
{\displaystyle f(x)}
with rational coefficients has
p
{\displaystyle p}
-integral coefficients if and only if
f
(
x
p
)
f
(
x
)
p
≡
1
mod
p
Z
p
[
[
x
]
]
.
{\displaystyle {\frac {f(x^{p})}{f(x)^{p}}}\equiv 1\mod p\mathbb {Z} _{p}[[x]].}
When
f
(
x
)
=
E
p
(
x
)
{\displaystyle f(x)=E_{p}(x)}
, we have
f
(
x
p
)
f
(
x
)
p
=
e
−
p
x
,
{\displaystyle {\frac {f(x^{p})}{f(x)^{p}}}=e^{-px},}
where the constant term is 1 and all higher coefficients are in
p
Z
p
{\displaystyle p\mathbb {Z} _{p}}
.
The public source identifies “Artin–Hasse exponential” as in mathematics, a power series named after Emil Artin and Helmut Hasse. This brief keeps that definition visible, then builds a research path around Artin, Hasse and exponential.
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