Let $q$ be a prime power. In function field arithmetics I frequently read about formal Laurent series $\mathbb{F}_q((\frac{1}{t}))$ and then it is said that $\frac{1}{t}$ is a "parameter at infinity".
What does this mean? What is the absolute value here?
I thought in $\mathbb{F}_q((t))$ one usually takes the following absolute value
$||\sum_{n=N}^\infty a_n t^n||= \lambda^{-N}$,
where $\lambda>1$ and $N$ is the smallest index $\in\mathbb{Z}$, such that $a_N\neq 0$. Since the zero series has no such value then $||0||=\lambda^{-\infty}=0$.
A local parameter at a point $c$ on a smooth projective curve $C$ is a rational function on $C$ that has a simple root at $c$. People call $t^{-1}$ a parameter at infinity because $\mathbb{F}_q(t)$ is the function field of the projective line $\mathbb{P}^1$ and $t^{-1}$ has a simple root at the point $\infty\in\mathbb{P}^1$.
The absolute value on $\mathbb{F}_q((t^{-1}))$ is given by $$ \left\|\sum_{n=N}^\infty a_n t^{-n}\right\|=\lambda^{-N},\hspace{10mm}a_N\neq 0, $$ where $\lambda>1$ is a fixed constant. Another way to describe the absolute value: for $f\in\mathbb{F}_q((t^{-1}))$ we take $\|f\|=\lambda^{-v_\infty(f)}$, where $v_{\infty}$ means order of vanishing at $\infty$.