Algebraic Geometry: Varieties & Elliptic Curves
Zero sets of polynomials, elliptic curves, and a hint of the algebra–geometry dictionary.
An affine variety over an algebraically closed field $k$ is the zero set
$$V(I) = \{ x \in k^n : f(x) = 0 \;\forall f \in I \}, \quad I \subseteq k[x_1, \ldots, x_n].$$Hilbert's Nullstellensatz establishes a bijection between radical ideals and affine varieties: $I(V(J)) = \sqrt J$ and $V(I(W)) = W$. Algebra ↔ geometry.
Projective varieties live in $\mathbb P^n = (k^{n+1} \setminus \{0\}) / \sim$ where $\lambda$-rescaling identifies points. Bezout's theorem: two projective plane curves of degrees $d, e$ without common components intersect in exactly $de$ points (counted with multiplicity, in $\mathbb P^2(\bar k)$).
Elliptic curves are smooth projective curves of genus 1, $y^2 = x^3 + a x + b$ with $\Delta = -16(4a^3 + 27b^2) \neq 0$. Their points form an abelian group via the chord-and-tangent law: $P + Q + R = 0$ when $P, Q, R$ are collinear. Mordell's theorem: $E(\mathbb Q)$ is finitely generated. Elliptic curves underlie BSD conjecture (Millennium Prize), modular forms, Wiles's proof of Fermat's Last Theorem, and modern ECC cryptography.