19 - Motivating Motives

Remember the formula:

number of points over \(\displaystyle{ \mathbb{F}_{p^n} \quad = \quad \sum_{k = 0}^{2d} \sum_{i = 1}^{\beta_k} \; (-1)^k \alpha_{ik}^n }\)

where \(\beta_k\) is the kth Betti number of the variety and \(|\alpha_{ik}| = p^{k/2}\).

Grothendieck's dream: we can always break the variety into abstract chunks called motives of dimension \(k = 0, 1, \dots, 2d\).

The k-dimensional motives contribute terms of the form \((-1)^k \, \alpha_{ik}^n\) to the number of points.

So, each chunk contributes to the number of points in our variety.
And it can contribute a negative number of points!