P-adic exponential function

Definition

The usual exponential function on 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 Cp, the completion of the algebraic closure of Qp, by :\exp_p(z)=\sum_{n=0}^\infty\frac{z^n}{n!}. However, unlike exp which converges on all of C, expp only converges on the disc :|z|_p 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 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 expp(x) does not converge at . It is possible to choose a number e to be a p-th root of expp(p) for p ≠ 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 Cp satisfying |x|p p(z) for |z − 1|p p(zw) = logp**z + logp**w. The function logp can be extended to all of C (the set of nonzero elements of Cp) by imposing that it continues to satisfy this last property and setting logp(p) = 0. Specifically, every element w of C can be written as w = pr·ζ·z with r a rational number, ζ a root of unity, and |z − 1|p p(w) = logp(z). This function on C is sometimes called the Iwasawa logarithm to emphasize the choice of logp(p) = 0. In fact, there is an extension of the logarithm from |z − 1|p p(p) in Cp.

Properties

If z and w are both in the radius of convergence for expp, then their sum is too and we have the usual addition formula: expp(z + w) = expp(z)expp(w).

Similarly if z and w are nonzero elements of Cp then logp(zw) = logp**z + logp**w.

For z in the domain of expp, we have expp(logp(1+z)) = 1+z and logp(expp(z)) = z.

The roots of the Iwasawa logarithm logp(z) are exactly the elements of Cp of the form pr·ζ where r is a rational number and ζ is a root of unity.

Note that there is no analogue in Cp of Euler's identity, e2πi = 1. This is a corollary of Strassmann's theorem.

Another major difference to the situation in C is that the domain of convergence of expp is much smaller than that of logp. A modified exponential function — the Artin–Hasse exponential — can be used instead which converges on |z|p

Notes

References

Citations

List of references

• Chapter 12 of
• {{Citation | author-link=Henri Cohen (number theorist)

References

1. {{harvnb. Robert. 2000
2. {{harvnb. Cohen. 2007
3. {{harvnb. Cohen. 2007
4. {{harvnb. Cohen. 2007