p-adic exponential function

In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the p-adic logarithm.

Definition

The usual exponential function on $\mathbb {C}$ is defined by the infinite series

$\exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.$

Entirely analogously, one defines the exponential function on $\mathbb {C} _{p}$, the completion of the algebraic closure of $\mathbb {Q} _{p}$, by

$\exp _{p}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.$

However, unlike exp which converges on all of $\mathbb {C}$, $\exp _{p}$ only converges on the disc

$|z|_{p}<p^{-1/(p-1)}.$

This is because p-adic series converge if and only if the summands tend to zero, and since the $n!$ in the denominator of each summand tends to make them large p-adically, a small value of z is needed in the numerator. It follows from Legendre's formula that if $|z|_{p}<p^{-1/(p-1)}$ then ${\frac {z^{n}}{n!}}$ tends to $0$, p-adically.

Although the p-adic exponential is sometimes denoted $e^{x}$, the number e itself has no p-adic analogue. This is because the power series $\exp _{p}(x)$ does not converge at $x=1$. It is possible to choose a number $e$ to be a p-th root of $\exp _{p}(p)$ for $p\neq 2$, but there are multiple such roots and there is no canonical choice among them.

p-adic logarithm function

The power series

$\log _{p}(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}},$

converges for $x$ in $\mathbb {C} _{p}$ satisfying $|x|_{p}<1$ and so defines the p-adic logarithm function $\log _{p}(z)$ for $|z-1|_{p}<1$ satisfying the usual property $\log _{p}(zw)=\log _{p}(z)+\log _{p}(w)$. The function $\log _{p}$ can be extended to all of $\mathbb {C}$×
p  (the set of nonzero elements of $\mathbb {C} _{p}$) by imposing that it continues to satisfy this last property and setting $\log _{p}(p)=0$. Specifically, every element $w$ of $\mathbb {C}$×
p  can be written as $w=p^{r}\cdot \zeta \cdot z$ with $r$ a rational number, $\zeta$ a root of unity, and $|z-1|_{p}<1$, in which case $\log _{p}(w)=\log _{p}(z)$. This function on $\mathbb {C}$×
p  is sometimes called the Iwasawa logarithm to emphasize the choice of $\log _{p}(p)=0$. In fact, there is an extension of the logarithm from $|z-1|_{p}<1$ to all of $\mathbb {C}$×
p  for each choice of $\log _{p}(p)$ in $\mathbb {C} _{p}$.

Properties

If $z$ and $w$ are both in the radius of convergence for $\exp _{p}$, then their sum is too and we have the usual addition formula: $\exp _{p}(z+w)=\exp _{p}(z)\exp _{p}(w)$.

Similarly if $z$ and $w$ are nonzero elements of $\mathbb {C} _{p}$ then $\log _{p}(zw)=\log _{p}(z)+\log _{p}(w)$.

For $z$ in the domain of $\exp _{p}$, we have $\exp _{p}(\log _{p}(1+z))=1+z$ and $\log _{p}(\exp _{p}(z))=z$.

The roots of the Iwasawa logarithm $\log _{p}(z)$ are exactly the elements of $\mathbb {C} _{p}$ of the form $p^{r}\cdot \zeta$ where $r$ is a rational number and $\zeta$ is a root of unity.

Note that there is no analogue in $\mathbb {C} _{p}$ of Euler's identity, $e^{2\pi i}=1$. This is a corollary of Strassmann's theorem.

Another major difference to the situation in $\mathbb {C}$ is that the domain of convergence of $\exp _{p}$ is much smaller than that of $\log _{p}$. A modified exponential function — the Artin–Hasse exponential — can be used instead which converges on $|z|_{p}<1$.

References

List of references

• Chapter 12 of Cassels, J. W. S. (1986). Local fields. London Mathematical Society Student Texts. Cambridge University Press. ISBN 0-521-31525-5.
• Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337
• Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, ISBN 0-387-98669-3

External links

• p-adic exponential and p-adic logarithm