Scheme Theory & Weak Mordell-Weil for Elliptic Curves Over Number Fields

Sammanfattning: We provide an introduction to scheme-theoretic algebraic geometry, which studies spaces that are in essence locally solutions to systems of polynomial equations, and prove the weak Mordell-Weil theorem. The weak Mordell-Weil theorem states that for an elliptic curve $E$ over a number field $K$, the quotient $E(K)/mE(K)$ is finite for all $m\geq 2$. The proof is adapted from a proof in the language of classical varieties, and uses some theorems from algebraic number theory (e.g. Hermite-Minkowski).