Given a d-dimensional smooth projective variety defined over the field with \(p\) elements,
its number of points over \(\mathbb{F}_{p^n}\) is $$ \sum_{k = 0}^{2d} \sum_{i = 1}^{\beta_k} \; (-1)^k \alpha_{ik}^n $$ where \(|\alpha_{ik}| = p^{k/2}\) and \(\beta_k\) is the kth "Betti number" of the variety:
that is, the rank of its kth cohomology group.

This conjecture is usually phrased in terms of a "zeta function". Then it claims this function has zeros and poles on the lines Re(z) = k/2: zeros when k is odd and poles when k is even.