Why exponential function on p-adic numbers is meaningless?

Question

In the notes, page 3, it is said that $e^{2\pi i r y}$ is meaningless if $y$ is a general p-adic number. Why exponential function on p-adic numbers is meaningless? Thank you very much.

Answer 1

It’s not the exponential function for $p$-adic numbers that’s meaningless; rather it’s the act of multiplying the real number $\pi$ by a nonrational $p$-adic number that’s meaningless. There’s no way of multiplying a real times a $p$-adic unless one of them is rational.

On the other hand, there is a $p$-adic exponential function, but it has nothing to do with the case that Conrad is discussing in these notes. It’s defined by the same power series that you learned in Calculus, but when considered as a function on a $p$-adic domain (whether $\mathbb Q_p$ or a complete field extension of $\mathbb Q_p$), its domain of definition is lamentably small, that is $\exp(z)$ converges at $z$ only when $v_p(z)>1/(p-1)$, where $v_p$ is the additive $p$-adic valuation normalized so that $v_p(p)=1$. In the language of absolute values, you need $|z|_p<\bigl(|p|_p\bigr)^{1/(p-1)}$. In particular, you can’t speak of $\exp(1)$, which would be, if it existed, the $p$-adic number corresponding to $e$.