13 - Motivating Motives

The simplest example: the projective line $\mathbb{P}^1$.
Over the field with q = pⁿ elements this has $$ q + 1 $$ points.

It's easy to split $\mathbb{P}^1$ into two parts: a copy of $\mathbb{F}_q$, with q points, and a point at infinity.

Over the complex numbers $\mathbb{P}^1$ looks like this: