produce primitive p-th root of unity

Dwork is one of the mathematicians that I admire so much.

Recently I learned how he produced primitive p-th roots of unity in \mathbb{C}_p “by hand”, which is interesting enough for me to write it down here.

Let \pi \in \mathbb{C}_p satisfies \pi^{p-1} = -p. Consider the following function \theta(z) = Exp(\pi z - \pi z^p). Then after plugging \{1, \zeta, \zeta^2, \ldots, \zeta^{p-2}\} in \theta(z), we will miraculously get all the primitive p-th roots of unity in \mathbb{C}_p. Here \zeta is a primitive (p-1)-st root of unity. The function \theta(z) bears the name “Dwork’s exponential”, I believe.

To prove the above statement, it suffices to prove it for z=1. So we need to prove \theta(1)^p = 1 yet \theta(1) \not= 1.

One thing to notice at the first place is that the convergence range of exponential function is the open disc of radius |\pi|, and the polynomial \pi z - \pi z^p has norm exactly |\pi| on the closed disc of radius 1. So the value \theta(1) cannot be computed by plugging 1 into the polynomial, which is 0, and then plug the output in the exponential function.

However, \theta(z)^p = Exp(\pi z - \pi z^p)^p = Exp(p\pi z - p \pi z^p). Now the polynomial p (\pi z - \pi z^p) has smaller norm on the closed disc, therefore we may compute the value at 1 by means in the previous paragraph. Hence we see that \theta(1)^p = 1.

Now we need to see that \theta(1) \not= 1. In the following we will show that \theta(1) - 1 = \pi \mod(\pi^2). To achieve this, let us rewrite \theta(z) = E_p(z) F(z). Here E_p(z) = Exp(x + x^p/p + x^{p^2}/p^2 + \ldots), and F(z) = Exp(-x^{p^2}/p^2 - x^{p^3}/p^3 - \ldots), where x = \pi z. The function E_p(z) is called the Artin-Hasse exponential. There’s a lemma of Dwork telling us that E_p(z) \in \mathbb{Z}_p[[\pi z]], hence E_p(1) = 1 + \pi \mod (\pi^2). On the other hand it’s easy to show directly that F(1) = 1 \mod(\pi^2). Put everything together, we see that \theta(1) - 1 = (E_p(1) F(1)) - 1 = (1+\pi) - 1 = \pi \mod(\pi^2) which is what we want to show.

In fact, before doing everything we probably have to show the convergence radius of \theta(z) is strictly bigger than 1. This actually easily follows from the estimate of E_p(z) \text{ and } F(z).

From the above construction, we see that there is a bijection between sets from (p-1)-st roots of -p and primitive p-th roots of unity. This is of course well known, since if we let w = \zeta_p - 1, then we’ll have w^{p-1} = -(p + \binom{p}{2} w + \ldots + p w^{p-2}).

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