9 - Motivating Motives

Since \(y^2 + y = x^3 + x \) is one equation with two unknowns
we might naively guess that over the field with pⁿ elements it has pⁿ solutions.

This is pretty close! This is the "main term". Let's subtract it off:

n pⁿ number of solutions number of solutions - pⁿ
1 2 4 2

2 4 4 0

3 8 4 -4

4 16 24 8

5 32 24 -8

6 64 64 0

7 128 144 16

8 256 224 -32

9 512 544 32

10 1024 1024 0

n	pⁿ	number of solutions	number of solutions - pⁿ
1	2	4	2
2	4	4	0
3	8	4	-4
4	16	24	8
5	32	24	-8
6	64	64	0
7	128	144	16
8	256	224	-32
9	512	544	32
10	1024	1024	0

We see oscillatory corrections — growing exponentially, but at a rate slower than pⁿ.