M2R Group 18

This is my 2022 M2R group project. Elliptic curves are smooth curves of genus one with a fixed rational point. From a relatively elementary point of view, they are defined by Weierstrass equations. The group law on elliptic curves can be constructed in two distinct ways, either geometrically or abstractly. We provided a detailed proof of the Mordell's theorem, and some in-depth discussion of elliptic curves in cryptography. This report also includes an introductory exposition to the background and use of Riemann-Roch on elliptic curves. With the help of this powerful algebraic geometry machine, the group was able to construct isogenies and Tate modules on elliptic curves and studied their properties. After obtaining the rejoiceful facts about elliptic curves and the endomorphisms on them, we would prove the Hasse bound and Weil conjectures for elliptic curves, from which we obtain the main result of this project --- the p-Riemann Hypothesis.