I am current studying Arithmetic Geometry, and I would like to know any sources containing a proof of Weil's generalization of Hasse's Theorem on elliptic curves over finite field. I don't think Silverman contain a proof of this result. The result I am looking for is the following: Let $E$ be a curve of genus $g$ over a finite field $\mathbb{F}_{q^m}$, where $q$ is a prime, then $|E(\mathbb{F}_{q^m})-q^m-1|\leq 2gq^{m/2}$
I know Tao has a proof using more elementary methods, which I am not interested in. Instead, I would like an arithemtic algebraic geometric proof. My backgrounds are the first 4 chapters of Hartshorne, and the first 5 chapters of Silverman's Arithmetic of Elliptic Curves. Thanks in advance.
Please edit the question to clarify what "Weil's generalization" refers to: do you mean the Riemann hypothesis for curves over finite fields? If so, see Section 3.3 in Mustata's notes here.