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.

The usual exponential function on C is defined by the infinite series

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

Entirely analogously, one defines the exponential function on C_p, the completion of the algebraic closure of Q_p, by

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

However, unlike exp which converges on all of C, exp_p only converges on the disc

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

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 ${\displaystyle |z|_{p}<p^{-1/(p-1)}}$ then ${\displaystyle {\frac {z^{n}}{n!}}}$ tends to ${\displaystyle 0}$, 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 ≠ 2,^[a] but there are multiple such roots and there is no canonical choice among them.^[1]

The power series

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

converges for x in 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_pz + log_pw. The function log_p can be extended to all of C ×
p  (the set of nonzero elements of C_p) by imposing that it continues to satisfy this last property and setting log_p(p) = 0. Specifically, every element w of C ×
p  can be written as w = p^r·ζ·z with r a rational number, ζ a root of unity, and |z − 1|_p < 1,^[2] in which case log_p(w) = log_p(z).^[b] This function on 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 C ×
p  for each choice of log_p(p) in C_p.^[3]

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 C_p then log_p(zw) = log_pz + log_pw.

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 C_p of the form p^r·ζ where r is a rational number and ζ is a root of unity.^[4]

Note that there is no analogue in C_p of Euler's identity, e^2πi = 1. This is a corollary of Strassmann's theorem.

Another major difference to the situation in 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.

• 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

• p-adic exponential and p-adic logarithm