All Questions

Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have: Fact ...

Let $K = \mathbb{F}_p$ be a field of positive characteristic $p > 0$. Consider a surface in $\mathbb{A}^3_K$ of the following form $$ S = \{f_1(x_0)y_0^2+f_2(x_0)y_0y_1+f_3(x_0)y_0+f_4(x_0)y_1^2+...

I wish to ask the naive question: if we write down a random curve $C$ over $\mathbb{F}_q$, what can be said about the probability that $C(\mathbb{F}_q)=\emptyset$? Of course, this depends on the ...

Let $A \colon V \to W$ be a surjective linear map (defined over $\mathbb{Z}$), inducing a projection $\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$. Let $X \subseteq \mathbb{P}(V)$ and $Y \...

Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...