Can I define a morphism between complex numbers and $\mathbb{Q}_2$ in this way using Gelfond's constant?
I'm exploring a (morphism? perhaps not the right word?) between the 2-adic and complex numbers using Gelfond's constant and trying to understand whether what I'm doing makes sense.
If we define Gelfond's constant to be $X=(-1)^{-i}$
Then we might say this is the multivalued function taking values $\{e^{-\pi},e^{\pi},e^{3\pi},e^{5\pi},\ldots\}$
So we might consider the natural log of this divided by $\pi$ to be the odd numbers:
$X=(-1)^{-i}=\{\ldots e^{-\pi},e^{\pi},e^{3\pi},e^{5\pi},\ldots\}$
$$\frac{\ln(X)}{\pi}=2\mathbb{Z}-1$$
I think I'm on fairly solid ground so far - am I not?
Am I right to call this $\mathbb{N}_2^{\times}$?
It would seem $\displaystyle2\mathbb{Z}=\frac{\ln(X\cdot e^{\pi})}{\pi}=\frac{\ln(X)}{\pi}+1$
So I can define $\displaystyle\mathbb{Z}=\frac{\ln(X)}{2\pi}+\frac{1}{2}$
How might I now define the sets
A: $\langle2\rangle=\{\ldots1/2,1,2,4,8,\ldots\}$
B: $\mathbb{Z}_2$
C: $\mathbb{Z}_2^{\times}$
in a comparable way? And
D: the metric $\lvert\cdot\rvert_2$
E: the valuation $v_2(x)$
For C it would seem I can say $$\mathbb{Z}_2^{\times}\cap\mathbb{Q}=\ln\left(\frac{X^{\tfrac{\pi}{\ln(X)}}}{\pi}\right)$$ since this gives us the set of ratios of any two odd numbers.
For A, the best I can do is: $\langle2\rangle=2^{\mathbb{Z}}=\sqrt{2}X^{\tfrac{\ln{2}}{2\pi}}$
which again is a bit convoluted.
As for the metric; perhaps this doesn't have a place here if I'm only working with sets of numbers. Or maybe it can be a multi-valued metric. Not really sure where to start with that.
As a little background, the motivation behind this is to understand the orthogonality of the orbits of a certain two functions in $\mathbb{Z}_2$ and I'm investigating whether such a morphism will give me some extra tools like regular calculus and complex number algebra; with which I'm a little more familiar than p-adic calculus.