Posters: your group’s poster plan is due on canvas; check the Posters tab for the date and details, and make contact with your group and make a plan.

Lecture notes: my notes from today’s lecture on projective space are posted (also on the Notes/Videos page and the Archive).

Solutions: solutions to the daily post due October 12th (elliptic curve addition) are posted. All daily post solutions are collected on the Archive page.

Relevant Textbook Sections:

5.3 (Projective space).

To Do:

1. Make sure you did last day’s daily post and compare to the solutions. 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.
2. Give an example of four different vectors in $\mathbb{R}^2$ which become the same in $\mathbb{P}^1_\mathbb{R}$.
3. In the textbook (Section 5.3) and in lecture, we studied $\mathbb{P}_{\mathbb{F}_3}^2$: the $26$ nonzero vectors in $\mathbb{F}_3^3$ fall into equivalence classes, each drawn in a different colour. Pick one colour, write out the vectors in that equivalence class, and check that they are equivalent. How many classes are there in total?
4. Reward yourself with this video.
5. 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.)
6. 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)?
7. 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.)
8. In lecture we saw that the elliptic curve $E: Y^2 Z = X^3 + aXZ^2 + bZ^3$ has exactly one point on the line at $\infty$, namely $[0,1,0] = \infty$. Check that the vertical line $x = c$ in the affine plane, written projectively as $X = cZ$, also passes through $[0,1,0]$. This is the precise sense in which “a vertical line meets $E$ at $\infty$,” which is what makes $P + (-P) = \infty$ in the group law from last week.