Geometry and Topology without figures

This blog serves as my preparation for my talk at Bloch's seminar workshop on Deligne-Lusztig varieties. I will present a talk about $\ell$-adic cohomology of Drinfeld curves, which are easiest examples of Degline-Lusztig varieties. To begin with, the story dates back to representation theory of finite group of Lie type (by a finite group of Lie type, I mean $G^F$ with $G$ a reductive group over a finite field and $F$ the Frobenious endomorphism). To say why these groups are important, it is the classification of finite simple groups saying that any finite simplegroup belongs to four classes: cyclic, alternating, $26$ sporadic groups, and the others are all of Lie type.  The central object of this blog is $SL_2(\mathbb{F}_q)$ with $q=p^r$ a prime power. It is a finite group of Lie type whose cardinality is $q(q-1)(q+1)$ (it's a fun exercise to show this in case you do not know). Not only $SL_2(\mathbb{F}_q)$, but the representation theory of any finite group over complex numbe...

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 (...