Concrete counting points on elliptic curves of the form $y^2 = x^3 + D$
In this post, I present explicit computations with elliptic curves of form $y^2=x^3+D$ (similar trick leads to curves $y^2 = x^3 + Dx$; the common point is they have complex multiplication). This was my TA for Ariane Mezard. at the Summer School on Galois Representations and Reciprocity last summer. The talk is based on the master thesis of Matteo Tamiozzo. Reminder on zeta functions For a prime $p$ and let $f(x,y,z) \in \mathbb{F}_p[x,y,z]$ be a homogeneous polynomial so that $$C = V(f) = \left \{[x:y:z] \in \mathbb{P}^2_{\mathbb{F}_p} \mid f(x,y,z) = 0 \right \}$$ is a smooth, projective curve. We define $$N_m = N_m(f) = \left \{P \ \text{has coordinates in} \ \mathbb{F}_{p^m} \right \}$$ to be the cardinality of solutions defined over the $\mathbb{F}_{p^m}$. We package these numbers $N_1,N_2,...,N_m,...$ into a zeta function $$Z_C(t) = \exp \left(\sum_{m=1}^{\infty} \frac{N_m}{m}t^m \right).$$ The celebrated result due to Weil and the school of Grothendieck gives us the (...