An elliptic curve is a nonsingular cubic equipped with a rational base point. Geometry turns its points into an abelian group, and arithmetic asks how that group behaves over \(\mathbb Q\), finite fields, and local fields.
Weierstrass equations and nonsingularity
Definition. Over a field of characteristic not \(2,3\), an elliptic curve has a Weierstrass model \[ E:y^2=x^3+Ax+B,\qquad \Delta=-16(4A^3+27B^2)\ne0, \] together with the point at infinity \(O\).
The discriminant condition says the cubic and its derivative have no common root, so the projective curve is nonsingular. Genus one plus the chosen point characterizes elliptic curves intrinsically.
The chord-tangent group law
Theorem. The points \(E(K)\) form an abelian group. If \(P=(x_1,y_1)\), \(Q=(x_2,y_2)\), \(P\ne\pm Q\), then \[ \lambda=\frac{y_2-y_1}{x_2-x_1},\qquad x_3=\lambda^2-x_1-x_2,\qquad y_3=\lambda(x_1-x_3)-y_1 \] give \(P+Q=(x_3,y_3)\). For doubling, \[ \lambda=\frac{3x_1^2+A}{2y_1}. \]
A line intersects a cubic in three points counted with multiplicity. Define \(P+Q\) as the reflection of the third intersection across the \(x\)-axis. Commutativity is geometric; associativity is most cleanly proved using divisor classes: \[ E(K)\cong\operatorname{Pic}^0(E),\qquad P\mapsto[(P)-(O)]. \]
Example. On \(E:y^2=x^3-2\), \(P=(3,5)\). The tangent slope is \(\lambda=27/10\), so \[ x(2P)=\frac{129}{100},\qquad y(2P)=-\frac{383}{1000}. \] Rational input produces rational output, but denominators grow rapidly.
Rational points
Theorem (Mordell-Weil). \[ E(\mathbb Q)\cong E(\mathbb Q)_{\mathrm{tors}}\oplus\mathbb Z^r \] for a finite torsion group and a finite rank \(r\).
The proof uses descent. A naive height \(h(x(P))\) measures arithmetic complexity, while the canonical height satisfies \[ \widehat h(nP)=n^2\widehat h(P),\qquad \widehat h(P)=0\Longleftrightarrow P\text{ is torsion}. \] Weak Mordell-Weil finiteness of \(E(\mathbb Q)/mE(\mathbb Q)\), together with height descent, yields finite generation.
Reduction modulo primes
At a prime \(p\nmid\Delta\), reduction gives an elliptic curve over \(\mathbb F_p\). Hasse's theorem states \[ \bigl|\#E(\mathbb F_p)-(p+1)\bigr|\le2\sqrt p. \] Writing \(a_p=p+1-\#E(\mathbb F_p)\), the local Euler factor is \[ 1-a_pp^{-s}+p^{1-2s}. \]
Local-global principles and descent
A rational point gives points over \(\mathbb R\) and every \(\mathbb Q_p\), but the converse can fail for genus-one curves. The Tate-Shafarevich group measures this failure. Selmer groups are finite, computable approximations that bound the Mordell-Weil rank.
Modularity
Theorem (modularity). Every elliptic curve over \(\mathbb Q\) is modular: its \(L\)-function agrees with the \(L\)-function of a weight-two newform of matching level.
This bridge between algebraic geometry and analysis supplied the route to Fermat's last theorem and motivates the Birch-Swinnerton-Dyer conjecture, which predicts \[ \operatorname{ord}_{s=1}L(E,s)=\operatorname{rank}E(\mathbb Q). \]