Your group Poster Plan is due on Wednesday on canvas; please make contact with your group and make a plan. Click “Posters” above for all the details.

Make sure you did last day’s daily post and compare to solutions (link here). Those computations are important; we’ll use them more. Notice in particular what to do if your slope has a denominator, or you are adding a point to itself.

Give an example of four different vectors in $\mathbb{R}^2$ which become the same in $\mathbb{P}^1_\mathbb{R}$.

In the notes, we studied $\mathbb{P}_{\mathbb{F}_3}^2$, showing 26 equivalence classes, each a different colour. Pick one colour, write out the vectors in that equivalence class, and check that they are equivalent.

Compute a full list of the elements in $\mathbb{P}_{\mathbb{F}_5}^1$. Each element is an equivalence class (a bunch of different vectors that differ by scalar multiplication), so write out all the vectors in each equivalence class. (We did this for $\mathbb{P}_{\mathbb{F}_3}^1$ in class.)

In general, in $\mathbb{P}_{\mathbb{F}_{p}}^k$, how many vectors are in each equivalence class? For example, in the example from lecture, in $\mathbb{P}_{\mathbb{F}_{3}}^1$, there were 2 in each class. What is the size of $\mathbb{P}_{\mathbb{F}_{p}}^k$ (i.e. how many equivalence classes)?

Find all the points of the affine equation $y^2 = xy – 1$ on the “line at $\infty$”. (Hint: we did this with the elliptic curve at the end of lecture: first homogenize the equation so all terms have the same degree, then break into $Z=0$ and $Z \neq 0$ cases.)