Elliptic curves

Intuition [Beginner]

An elliptic curve is a smooth curve of genus 1 with one marked point. Concretely, over the rationals or any field where $2$ and $3$ are invertible, the equation $y^2=x^3+ax+b$ defines such a curve in the plane, provided the cubic has no double root. Add the point at infinity and you have an elliptic curve; the marked point is that point at infinity.

What makes elliptic curves remarkable: the points form a group. Pick two points $P$ and $Q$ on the curve, draw the straight line through them, and that line crosses the curve at exactly one more point $R$. Reflect $R$ across the $x$-axis and call the result $P+Q$. This recipe is symmetric, has the point at infinity as identity, and is associative. The group law is built from the geometry of the cubic — three points sum to zero exactly when they are collinear.

The point group $E(\mathbb{Q})$ of rational solutions sits inside the curve as a finitely generated abelian group: a finite torsion part plus a free ${\mathbb{Z}}^r$ for some non-negative integer $r$ called the rank. Mordell proved this in 1922 — the deep arithmetic statement that there are only finitely many "essential" rational points from which every other rational point is built.

Visual [Beginner]

The cubic curve $y^2=x^3-x$ in the real plane: a connected oval on the right and a closed loop on the left, meeting the $x$-axis at $-1,0,1$. Two marked points $P,Q$ on the curve, the line through them crossing the curve at a third point $R^{\prime }$, and the reflection of $R^{\prime }$ across the $x$-axis giving $P+Q$.

The picture records the chord-and-tangent group law: three collinear points sum to zero, with the convention that the point at infinity plays the role of the identity element.

Worked example [Beginner]

Take $E:y^2=x^3-x$ over $\mathbb{Q}$. The cubic $x^3-x=x(x-1)(x+1)$ has three distinct roots, so the curve is smooth. The discriminant is $\Delta =-16\cdot 4\cdot (-1{)}^3=64\ne 0$, confirming smoothness.

The three points where the curve meets the $x$-axis are $P=(0,0)$, $Q=(1,0)$, $S=(-1,0)$. The horizontal line $y=0$ meets the curve at exactly these three points and no other. By the chord rule, $P+Q+S=O$ (the point at infinity), so $P+Q=-S$. The negation rule on this curve sends $(x,y)$ to $(x,-y)$, and since $S$ has $y=0$, $-S=S$. Therefore $P+Q=S=(-1,0)$.

Each of $P,Q,S$ has order 2, since adding any of them to itself uses the tangent line, which is vertical at a point with $y=0$, and the third intersection of a vertical line with the curve is the point at infinity, so $2P=-O=O$. Together with the point at infinity, the points $\{O,P,Q,S\}$ form a group of order 4 with every non-identity element of order 2, namely the Klein four-group $(\mathbb{Z}/2{)}^2$.

What this tells us: $E(\mathbb{Q}{)}_{\mathrm{tors}}\supseteq (\mathbb{Z}/2{)}^2$ for this curve, and a closer analysis shows $E(\mathbb{Q}{)}_{\mathrm{tors}}=(\mathbb{Z}/2{)}^2$ — the rational torsion is exactly these four points.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $k$ be a field. An elliptic curve over $k$ is a pair $(E,O)$ where $E$ is a smooth projective geometrically connected curve over $k$ of genus 1 and $O\in E(k)$ is a $k$-rational point. Equivalently, $E$ is a 1-dimensional abelian variety over $k$ — a connected smooth projective group scheme over $k$ of dimension 1, with $O$ the identity element.

Weierstrass model. Assume $\mathrm{char}k\ne 2,3$ for the simplest setup. Riemann-Roch on the genus-1 curve $E$ gives $\dim H^0(E,\mathcal{O}(nO))=n$ for $n\ge 1$ 04.04.01. The basis $\{1,x\}$ of $H^0(E,\mathcal{O}(2O))$ together with $y\in H^0(E,\mathcal{O}(3O))$ chosen so that $\{1,x,y\}$ spans $H^0(E,\mathcal{O}(3O))$ embeds $E$ in ${\mathbb{P}}^2$ as the projective closure of the affine cubic $$ y^2 = x^3 + a x + b, \qquad a, b \in k, $$ with $O$ identified as the unique point at infinity $[0:1:0]$. The discriminant of the model is $\Delta =-16(4a^3+27b^2)$; smoothness of $E$ is equivalent to $\Delta \ne 0$, equivalently the cubic $x^3+ax+b$ has three distinct roots in $\bar{k}$. The $j$-invariant is $$ j(E) = -1728 \cdot \frac{(4a)^3}{\Delta} = 1728 \cdot \frac{4 a^3}{4 a^3 + 27 b^2}. $$ Two elliptic curves over an algebraically closed field $\bar{k}$ are isomorphic iff they have the same $j$-invariant, and every $j_0\in \bar{k}$ occurs as $j(E)$ for some elliptic curve $E/\bar{k}$. The $j$-invariant therefore identifies the moduli space of elliptic curves over $\bar{k}$ with the affine line ${\mathbb{A}}_{\bar{k}}^1$.

Definition (group law). The group law on $E$ is the unique morphism $\mu :E\times E\to E$ such that $\mu (O,P)=P$, the inverse $\iota :E\to E$ the unique morphism such that $\mu (P,\iota (P))=O$, and three points $P,Q,R\in E$ with $P+Q+R=O$ iff they are collinear in the plane embedding (counted with multiplicity at points where the line is tangent to $E$). Equivalently, the map $E(k)\to {\mathrm{Pic}}^0(E)$, $P\mapsto [P-O]$, is a group isomorphism — Riemann-Roch on the genus-1 curve $E$ gives $\dim H^0(E,\mathcal{O}(D))=\mathrm{deg}D$ for a divisor $D$ of degree $\ge 1$, so every degree-zero divisor class on $E$ has a unique representative of the form $P-O$ with $P\in E$.

Sign convention. The negation on the Weierstrass model sends $(x,y)$ to $(x,-y)$; the point at infinity is fixed. The group law is commutative, with identity $O$. The Mordell-Weil rank of $E$ over a number field $K$ is $r={\dim }_{\mathbb{Q}}E(K){\otimes }_{\mathbb{Z}}\mathbb{Q}$, the rank of the free part of the finitely generated abelian group $E(K)$.

Counterexamples to common slips.

• Genus-1 without rational point. A smooth projective curve of genus 1 over $k$ need not be an elliptic curve: it must come equipped with a $k$-rational point. Over $\mathbb{Q}$, the curve $3x^3+4y^3+5z^3=0$ in ${\mathbb{P}}^2$ (Selmer's curve) has genus 1 but no $\mathbb{Q}$-rational points; it is a principal homogeneous space for an elliptic curve, not itself an elliptic curve.
• Singular Weierstrass cubics. The equation $y^2=x^3$ defines a curve with a cusp at the origin; the equation $y^2=x^3+x^2$ defines a curve with a node. Neither is an elliptic curve in the strict sense — both have $\Delta =0$. Their smooth loci do carry group structures (the additive group ${\mathbb{G}}_a$ and multiplicative group ${\mathbb{G}}_m$ respectively), but these are generalised elliptic curves, not elliptic curves.
• Characteristic 2 and 3. Outside the assumption $\mathrm{char}k\ne 2,3$, the short Weierstrass form is insufficient: one needs the long Weierstrass form $y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6$ with five coefficients, and the discriminant formula is correspondingly more complicated.
• Confusing $j$-invariant with isomorphism over $k$. Over a non-algebraically-closed field $k$, two elliptic curves with the same $j$-invariant are twists of each other but need not be isomorphic over $k$. The set of $k$-twists of $E$ is parametrised by $H^1(\mathrm{Gal}(\bar{k}/k),\mathrm{Aut}(E_{\bar{k}}))$.

Key theorem with proof [Intermediate+]

Theorem (Mordell-Weil, 1922). Let $E$ be an elliptic curve over $\mathbb{Q}$. The group $E(\mathbb{Q})$ is finitely generated.

Proof. The argument has two ingredients: weak Mordell-Weil (the finite-quotient statement $E(\mathbb{Q})/nE(\mathbb{Q})$ is finite for some integer $n\ge 2$), and the theory of heights (a positive function $h:E(\mathbb{Q})\to {\mathbb{R}}_{\ge 0}$ controlled by the group law).

Step 1 — Weak Mordell-Weil. Take $n=2$ for concreteness; the proof for general $n\ge 2$ is parallel. The multiplication-by-$2$ map $[2]:E\to E$ is a finite étale cover of degree 4 over the open complement of $E[2]$. The Kummer sequence $0\to E[2]\to E\stackrel{[2]}{\to }E\to 0$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-modules gives a long exact cohomology sequence with connecting map $$ \delta : E(\mathbb{Q}) / 2 E(\mathbb{Q}) \hookrightarrow H^1(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), E[2]). $$ The image of $\delta$ lies in the subgroup of cohomology classes unramified outside the finite set $S$ of primes of bad reduction together with $\{2,\infty \}$. Hermite-Minkowski / class-number finiteness for $\mathbb{Q}$ then forces this subgroup to be finite (it is ${\mathrm{Sel}}_2(E/\mathbb{Q})$, the 2-Selmer group, contained in a finite group of cohomology classes ramified only at $S$). Therefore $E(\mathbb{Q})/2E(\mathbb{Q})$ is finite.

Step 2 — Heights. The naive height of a rational point $P=(x,y)\in E(\mathbb{Q})$ written in lowest terms with $x=a/b$, $\gcd (a,b)=1$, $b>0$, is $H(P):=\max (|a|,|b|)$, and the logarithmic naive height is $h(P):=\log H(P)$ (with $h(O)=0$). The Néron-Tate canonical height is the limit $$ \hat h(P) := \lim_{n \to \infty} \frac{h(2^n P)}{4^n}, $$ which exists by a Cauchy-sequence argument and satisfies the parallelogram law $$ \hat h(P + Q) + \hat h(P - Q) = 2 \hat h(P) + 2 \hat h(Q). $$ The parallelogram law implies $\hat{h}$ is a quadratic form on $E(\mathbb{Q})\otimes \mathbb{ZR}$, and the associated bilinear pairing $⟨P,Q⟩:=\frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))$ is positive-definite on $E(\mathbb{Q})\otimes \mathbb{Q}$ modulo torsion. Two further properties hold:

(i) For every constant $C>0$, the set $\{P\in E(\mathbb{Q}):\hat{h}(P)\le C\}$ is finite. This is because $\hat{h}$ differs from $h$ by an $O(1)$ term, and there are only finitely many rational points of bounded naive height.

(ii) $\hat{h}(nP)=n^2\hat{h}(P)$ for every $n\in \mathbb{Z}$.

Step 3 — Descent. Take a complete set of representatives $Q_1,\dots ,Q_m$ for $E(\mathbb{Q})/2E(\mathbb{Q})$ (finite by Step 1). Set $C:={\max }_i\hat{h}(Q_i)$. Let $P\in E(\mathbb{Q})$. Write $P=Q_{i_0}+2P_1$ for some $i_0$ and $P_1\in E(\mathbb{Q})$, then $P_1=Q_{i_1}+2P_2$, and so on. The parallelogram law gives $$ 4 \hat h(P_1) = \hat h(2 P_1) = \hat h(P - Q_{i_0}) \leq 2 \hat h(P) + 2 \hat h(Q_{i_0}) - \hat h(P + Q_{i_0}) \leq 2 \hat h(P) + 2 C, $$ hence $\hat{h}(P_1)\le \frac{1}{2}\hat{h}(P)+\frac{1}{2}C$. Iterating, $\hat{h}(P_n)\le 2^{-n}\hat{h}(P)+C$, so for some $N$ we have $\hat{h}(P_N)\le 2C$. By (i) the set $S:=\{Q\in E(\mathbb{Q}):\hat{h}(Q)\le 2C\}$ is finite. Backtracking, $P$ lies in the subgroup of $E(\mathbb{Q})$ generated by $\{Q_1,\dots ,Q_m\}\cup S$, a finite set. Therefore $E(\mathbb{Q})$ is finitely generated. $\square$

The descent argument is Fermat's infinite descent in the form Mordell adapted: every rational point reduces, after finitely many halvings, to a point of bounded height, and bounded-height rational points are finite in number. Weil 1929 ^{[Weil 1929]} extended the proof to elliptic curves over arbitrary number fields by replacing $\mathbb{Q}$-class-number finiteness with $K$-class-number finiteness and the Dirichlet unit theorem.

Bridge. The Mordell-Weil structure $E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus E(\mathbb{Q}{)}_{\mathrm{tors}}$ here builds toward Mazur's torsion theorem [pending] (Mazur 1977 ^{[Mazur 1977]}) which classifies the 15 possible isomorphism types of $E(\mathbb{Q}{)}_{\mathrm{tors}}$, and toward the Birch and Swinnerton-Dyer conjecture which predicts the rank $r$ from the order of vanishing of the $L$-function at $s=1$. The same descent template appears again in the proof of finite generation for the Mordell-Weil group of an abelian variety over a number field, and the Néron-Tate height extends to a positive-definite quadratic form on the Mordell-Weil group of every abelian variety. The 2-Selmer group ${\mathrm{Sel}}_2(E/\mathbb{Q})$ that appears in Step 1 sits inside the Tate-Shafarevich group $\text{Sha}(E/\mathbb{Q})$, conjectured to be finite — a conjecture of Tate that connects to the BSD leading-term refinement and to the local-global principle for principal homogeneous spaces under $E$. The chord-and-tangent geometry of the group law is exactly what makes this descent computable: the formulas for $P+Q$ are rational functions in the coordinates of $P$ and $Q$, and the height behaviour under $[2]$ is the engine of the Step-3 argument. Putting these together, the theorem packages three apparently distinct ingredients — Galois cohomology (Selmer groups), Diophantine geometry (heights), and infinite descent — into a single finiteness statement, and its quantitative refinements form the modern arithmetic theory of elliptic curves.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none. Mathlib has the predicate EllipticCurve over a commutative ring with the discriminant non-zero condition (in Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass) and the chord-and-tangent group law on the affine model with point-at-infinity (in Mathlib.AlgebraicGeometry.EllipticCurve.Group). What is missing for the content of this unit is summarised in the lean_mathlib_gap field; the headline target — Mordell-Weil for $E(\mathbb{Q})$ — depends on Galois cohomology of $E[2]$, the Néron-Tate height construction, and Hermite-Minkowski / class-number finiteness, none currently formalised in the elliptic-curve API.

[object Promise]

Advanced results [Master]

Mazur's torsion theorem (Mazur 1977). For every elliptic curve $E$ over $\mathbb{Q}$, the torsion subgroup $E(\mathbb{Q}{)}_{\mathrm{tors}}$ is isomorphic to one of fifteen explicit groups: $\mathbb{Z}/n$ for $n\in \{1,2,3,4,5,6,7,8,9,10,12\}$, or $\mathbb{Z}/2\oplus \mathbb{Z}/2m$ for $m\in \{1,2,3,4\}$. All fifteen groups occur for some $E/\mathbb{Q}$. The proof goes via the moduli interpretation: a $\mathbb{Q}$-rational point on the modular curve $X_1(N)$ corresponds to an elliptic curve $E/\mathbb{Q}$ together with a $\mathbb{Q}$-rational point of order $N$. Mazur computed the rational points on $X_1(N)$ for the relevant range of $N$, showing that for $N\notin \{1,\dots ,10,12\}$ (with the natural extension to mixed torsion), $X_1(N)(\mathbb{Q})$ contains only cusps. The proof relies on Mazur's Eisenstein ideal analysis of the modular Jacobian $J_1(N)$, the determination of its $\mathbb{Q}$-rational torsion and its Mordell-Weil rank in terms of the Eisenstein quotient, and the Eichler-Shimura congruence relating Hecke operators to Frobenius.

Modularity theorem (Wiles 1995, Taylor-Wiles 1995, Breuil-Conrad-Diamond-Taylor 2001). Every elliptic curve $E$ over $\mathbb{Q}$ is modular: there exists a weight-2 cuspidal newform $f\in S_2({\Gamma }_0(N))$ of level $N=N(E)$ (the conductor of $E$) such that the $L$-function $L(E,s)$ equals the $L$-function $L(f,s)$ of the newform. Equivalently, $E$ admits a non-constant morphism $\varphi :X_0(N)\to E$ over $\mathbb{Q}$ for $N=N(E)$. Wiles proved the semistable case in 1995 ^{[Wiles 1995]} (sufficient for Fermat's Last Theorem via the Frey curve and the level-lowering theorem of Ribet), and the full theorem was completed by Breuil-Conrad-Diamond-Taylor in 2001 ^{[Breuil-Conrad-Diamond-Taylor 2001]}. The proof method is the modularity lifting theorem, $R=T$: identify a deformation ring $R$ parametrising suitable Galois representations with a Hecke algebra $T$ acting on modular forms.

BSD conjecture (Birch-Swinnerton-Dyer, 1965). The order of vanishing of $L(E,s)$ at $s=1$ equals the Mordell-Weil rank: ${\mathrm{ord}}_{s=1}L(E,s)=\mathrm{rank}E(\mathbb{Q})$. The leading-term refinement asserts $$ \lim_{s \to 1} \frac{L(E, s)}{(s - 1)^r} = \frac{#\Sha(E/\mathbb{Q}) \cdot \Omega_E \cdot R_E \cdot \prod_p c_p}{(#E(\mathbb{Q})_{\mathrm{tors}})^2}, $$ where $r$ is the rank, $\text{Sha}$ is the Tate-Shafarevich group, ${\Omega }_E$ is the real period, $R_E$ is the regulator (the determinant of the Néron-Tate height pairing on a basis of $E(\mathbb{Q})/\mathrm{tors}$), and $c_p$ are the local Tamagawa numbers. The conjecture is open in general; partial results include Coates-Wiles 1977 (rank 0 case for CM curves), Gross-Zagier 1986 / Kolyvagin 1989 (rank $\le 1$ for elliptic curves admitting a Heegner-point construction), and Kim 2007+ (extension via $p$-adic methods).

Hasse bound and Frobenius. Over a finite field ${\mathbb{F}}_q$ with $q=p^n$, the Frobenius endomorphism ${\varphi }_q:E\to E$, $(x,y)\mapsto (x^q,y^q)$, satisfies a quadratic relation ${\varphi }_q^2-a_q{\varphi }_q+q=0$ in the endomorphism ring $\mathrm{End}(E_{\overline{{\mathbb{F}}_q}})$, where $a_q=q+1-\#E({\mathbb{F}}_q)$ is the trace of Frobenius. The eigenvalues $\alpha ,\beta$ of ${\varphi }_q$ acting on the Tate module $T_{\ell }E$ for $\ell \ne p$ are complex conjugate algebraic integers with $|\alpha |=|\beta |=\sqrt{q}$, hence $|a_q|=|\alpha +\beta |\le 2\sqrt{q}$ — the Hasse bound. An elliptic curve $E/\overline{{\mathbb{F}}_p}$ is ordinary if $\#E[p](\overline{{\mathbb{F}}_p})=p$, supersingular if $\#E[p](\overline{{\mathbb{F}}_p})=1$ (equivalently, $a_p\equiv 0(\mathrm{mod}p)$ over ${\mathbb{F}}_p$). The endomorphism ring of an ordinary elliptic curve is an order in an imaginary quadratic field; that of a supersingular elliptic curve is an order in a quaternion algebra ramified at $p$ and $\infty$.

Tate's algorithm. For an elliptic curve $E$ over a complete discretely valued field $K$ (e.g. ${\mathbb{Q}}_p$) with valuation ring ${\mathcal{O}}_K$ and uniformiser $\pi$, the minimal Weierstrass model over ${\mathcal{O}}_K$ has reduction type at the residue field $k={\mathcal{O}}_K/\pi$ classified by Kodaira's symbols: $I_n$ (multiplicative, semistable, with $n$-component special fibre), $I_n^{\ast }$ ($n\ge 0$, additive after a quadratic twist), and the exceptional types $II,III,IV,I{I}^{\ast },II{I}^{\ast },I{V}^{\ast }$. Tate's algorithm 1975 Algorithm for determining the type of a singular fibre in an elliptic pencil computes the reduction type, the minimal model, and the local conductor exponent $f_p$ from the Weierstrass coefficients of any model.

CM theory. An elliptic curve $E$ over $\mathbb{C}$ has complex multiplication if $\mathrm{End}(E)\otimes \mathbb{Q}$ is a 2-dimensional $\mathbb{Q}$-algebra — necessarily an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$. Equivalently, $E\cong \mathbb{C}/\mathcal{O}$ for an order $\mathcal{O}$ in $K$. The main theorem of complex multiplication (Kronecker-Weber generalisation, Shimura-Taniyama 1961): the values of $j$ at CM points generate the Hilbert class field of $K$, and the values of suitable elliptic / Weber modular functions generate the ring class fields. CM elliptic curves over $\mathbb{Q}$ have $L$-functions equal to Hecke $L$-functions of associated $K$-Grössencharakters (Deuring 1953), making the modularity-of-CM-curves a classical theorem long predating Wiles.

Heegner points. For an elliptic curve $E/\mathbb{Q}$ of conductor $N$ and an imaginary quadratic field $K$ such that all primes dividing $N$ split in $K$ (the Heegner hypothesis), there is a systematic construction of $K$-rational points on $E$ via the modular parametrisation $\varphi :X_0(N)\to E$: take the image $\varphi (P_K)\in E(K)$ of a Heegner point $P_K\in X_0(N)$, where $P_K$ corresponds to a pair $(\mathbb{C}/{\mathcal{O}}_K,\mathbb{C}/{\mathfrak{n}}^{-1})$ with ${\mathcal{O}}_K$ the ring of integers and $\mathfrak{n}$ an ideal of norm $N$. Gross-Zagier 1986 Heegner points and derivatives of $L$-series showed that the trace down to $\mathbb{Q}$ of such Heegner points has Néron-Tate height equal to a non-zero multiple of $L^{\prime }(E,1)$ (the derivative at $s=1$). Combined with Kolyvagin's Euler system method, this proves the rank-$\le 1$ case of the BSD conjecture for elliptic curves of analytic rank $\le 1$.

Modular curves $X_0(N)$, $X_1(N)$ as moduli. The modular curve $Y_0(N)=\mathfrak{H}/{\Gamma }_0(N)$ parametrises pairs $(E,C)$ of an elliptic curve $E$ together with a cyclic subgroup $C\subset E$ of order $N$; the curve $Y_1(N)=\mathfrak{H}/{\Gamma }_1(N)$ parametrises pairs $(E,P)$ with $P\in E$ a point of order $N$. Their projective compactifications $X_0(N),X_1(N)$ add finitely many cusps, giving smooth projective curves over $\mathbb{Q}$. The $j$-invariant defines a holomorphic function on $\mathfrak{H}$ invariant under ${\mathrm{SL}}_2(\mathbb{Z})$, providing an isomorphism $j:Y(1):=\mathfrak{H}/{\mathrm{SL}}_2(\mathbb{Z})\cong {\mathbb{A}}_{\mathbb{C}}^1$, with $X(1)={\mathbb{P}}_{\mathbb{C}}^1$ obtained by adding the single cusp at $\infty$. The functional equation $j(-1/\tau )=j(\tau )$ exhibits $j$ as the foundational example of a modular function.

Synthesis. Elliptic curves are the simplest abelian varieties beyond the additive and multiplicative groups: 1-dimensional, projective, group-scheme-and-variety in one. The theory packages four apparently distinct ingredients into a single object. The first is the geometry: a smooth genus-1 curve with marked point, embedded in ${\mathbb{P}}^2$ as a Weierstrass cubic via Riemann-Roch 04.04.01, with its chord-and-tangent group law identified with ${\mathrm{Pic}}^0(E)$ on a genus-1 curve. The second is the arithmetic over number fields: the Mordell-Weil group $E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus \mathbb{Z}/n_1\oplus \mathbb{Z}/n_2$ is finitely generated (Mordell 1922 / Weil 1929), with torsion classified by Mazur 1977 and rank predicted by BSD via the $L$-function. The third is the automorphic side: the modularity theorem (Wiles-Taylor-BCDT) identifies $L(E,s)$ with the $L$-function of a weight-2 cuspidal newform on ${\Gamma }_0(N)$, building a bridge between geometry and analysis. The fourth is the moduli-theoretic side: $j$-invariant identifies the moduli of elliptic curves over $\bar{k}$ with ${\mathbb{A}}_{\bar{k}}^1$, modular curves $X_0(N),X_1(N)$ parametrise level structures, and CM points generate class fields of imaginary quadratic fields.

The same Néron-Tate height that powers Mordell-Weil reappears as the regulator in BSD; the same Galois cohomology that gives weak Mordell-Weil reappears in the Selmer-Shafarevich exact sequence; the same modular curves that classify torsion are the source of Heegner points used in Gross-Zagier-Kolyvagin. Putting these together, the foundational reason elliptic curves are the central object of modern arithmetic geometry is exactly that every major theorem about them connects to every other: rank, torsion, $L$-function, modularity, Heegner points, CM, reduction types, and Tate-Shafarevich — all are computed from the same Weierstrass coefficients $a,b$ via mutually-determining algorithms, and their interconnections are the engine of Diophantine geometry.

Full proof set [Master]

Theorem (Weak Mordell-Weil). Let $E$ be an elliptic curve over a number field $K$ and $n\ge 2$ an integer. Then $E(K)/nE(K)$ is finite.

Proof. The multiplication-by-$n$ map $[n]:E\to E$ is a finite étale isogeny of degree $n^2$ over the open complement of $E[n]$, with kernel $E[n]\cong (\mathbb{Z}/n{)}^2$ as a $\mathrm{Gal}(\bar{K}/K)$-module. The Kummer sequence $0\to E[n]\to E\stackrel{[n]}{\to }E\to 0$ as étale sheaves gives a long exact cohomology sequence; the connecting map yields the Kummer injection $$ \delta : E(K) / n E(K) \hookrightarrow H^1(\mathrm{Gal}(\bar K / K), E[n]). $$ The image lies in the $n$-Selmer group ${\mathrm{Sel}}_n(E/K)$, defined as the subgroup of cohomology classes that are unramified at all primes of good reduction and lie in the image of the local connecting maps for primes of bad reduction. By the standard finiteness argument — Hermite-Minkowski for $K$, plus finiteness of the class group $\mathrm{Cl}(K)$ and finite generation of the unit group ${\mathcal{O}}_K^{\times }$ — the group ${\mathrm{Sel}}_n(E/K)$ is finite ^{[Silverman §VIII]}. Therefore $E(K)/nE(K)\subseteq {\mathrm{Sel}}_n(E/K)$ is finite. $\square$

Theorem (Mordell-Weil). Statement and proof as in the Intermediate-tier Key theorem section, generalised to a number field $K$ in place of $\mathbb{Q}$.

Proof. The Intermediate-tier proof goes through with weak Mordell-Weil (above) replacing the $\mathbb{Q}$-specific Step 1 and the Néron-Tate height construction (existence and parallelogram law via the Tate limit, finite-many-points-of-bounded-height property) replacing the Step 2 ingredients. Step 3 (descent) is identical. $\square$

Theorem (Hasse bound). Let $E$ be an elliptic curve over ${\mathbb{F}}_q$. Then $|\#E({\mathbb{F}}_q)-q-1|\le 2\sqrt{q}$.

Proof sketch. The Frobenius endomorphism $\varphi :E\to E$, $(x,y)\mapsto (x^q,y^q)$, generates the kernel of the étale-cohomological action and satisfies a quadratic equation ${\varphi }^2-a_q\varphi +q=0$ over $\mathrm{End}(E_{\overline{{\mathbb{F}}_q}})$, where $a_q=q+1-\#E({\mathbb{F}}_q)$. The action of $\varphi$ on the $\ell$-adic Tate module $T_{\ell }E$ (for $\ell \ne p$) is given by a matrix in ${\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$ with characteristic polynomial $T^2-a_{q}T+q$. The two roots $\alpha ,\beta$ have $\alpha \beta =q$ and $\alpha +\beta =a_q$. The fact that $\varphi -n$ is an endomorphism for every integer $n$, with $\mathrm{deg}(\varphi -n)=(\varphi -n)(\varphi -n{)}^{\vee }=(\varphi -n)(\bar{\varphi }-n)=\varphi \bar{\varphi }-n(\varphi +\bar{\varphi })+n^2=q-n{a}_q+n^2\ge 0$ for all $n\in \mathbb{Z}$, forces the discriminant $a_q^2-4q\le 0$. Hence $|a_q|\le 2\sqrt{q}$. $\square$

Theorem (group law from ${\mathrm{Pic}}^0$). On a smooth projective genus-1 curve $E$ with marked point $O\in E(k)$, the map $E(k)\to {\mathrm{Pic}}^0(E)$, $P\mapsto [P-O]$, is a bijection. The induced group structure on $E(k)$ coincides with the chord-and-tangent law on a Weierstrass model.

Proof. Bijection: Riemann-Roch on a genus-1 curve and Serre duality, as in Exercise 5.

Group structure: a line $\ell :f=0$ in the plane meets the Weierstrass cubic $E\subset {\mathbb{P}}^2$ in a divisor $D_{\ell }$ of degree 3 (Bézout). Two lines $\ell ,{\ell }^{\prime }$ meeting $E$ in divisors $D_{\ell },D_{{\ell }^{\prime }}$ have $D_{\ell }{\sim }_{\mathrm{lin}}{D}_{{\ell }^{\prime }}$ (their difference is the divisor of the rational function $f/f^{\prime }$, hence principal). In particular, every $D_{\ell }$ is linearly equivalent to $3O$ (the divisor of any line through the inflection point at infinity). Three collinear points $P,Q,R$ on $E$ thus give $P+Q+R{\sim }_{\mathrm{lin}}3O$, i.e. $(P-O)+(Q-O)+(R-O){\sim }_{\mathrm{lin}}0$. Translating to ${\mathrm{Pic}}^0(E)$: $[P-O]+[Q-O]+[R-O]=0$. The chord rule on the Weierstrass model — three collinear points sum to zero in the group — therefore agrees with addition in ${\mathrm{Pic}}^0(E)$. $\square$

Theorem (Weierstrass embedding). Let $E$ be an elliptic curve over $k$ with $\mathrm{char}k\ne 2,3$. The complete linear system $|3O|$ defines a closed embedding ${\varphi }_{|3O|}:E\hookrightarrow {\mathbb{P}}^2$ with image cut out by an equation $y^2=x^3+ax+b$ for some $a,b\in k$.

Proof. By Riemann-Roch on the genus-1 curve $E$, $\dim H^0(E,\mathcal{O}(nO))=n$ for $n\ge 1$. Choose: a non-zero constant $1\in H^0(E,\mathcal{O})$, a function $x\in H^0(E,\mathcal{O}(2O))$ with a double pole at $O$ and no other poles (such $x$ exists because $\dim H^0(E,\mathcal{O}(2O))=2$ contains the constants and one new function), a function $y\in H^0(E,\mathcal{O}(3O))$ with a triple pole at $O$. The space $H^0(E,\mathcal{O}(6O))$ has dimension 6, but it contains the seven functions $1,x,y,x^2,xy,x^3,y^2$ (each with pole order at $O$ at most 6), so a $k$-linear relation $$ y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 $$ holds with $a_i\in k$. Comparing pole orders at $O$ shows the coefficients of $y^2$ and $x^3$ are non-zero; rescaling $y$ and $x$ absorbs them into 1. Substituting $y\mapsto y-(a_{1}x+a_3)/2$ and $x\mapsto x-a_2/3$ (valid because $\mathrm{char}k\ne 2,3$) eliminates the $a_1,a_2,a_3$ terms, giving the short Weierstrass form $y^2=x^3+ax+b$.

The map ${\varphi }_{|3O|}=[1:x:y]:E\to {\mathbb{P}}^2$ is well-defined on $E\setminus \{O\}$ (using the affine coordinates $(x,y)$) and sends $O$ to $[0:0:1]$ (using $y^{-1}$ as the local generator at $O$, giving $[y^{-1}:x{y}^{-1}:1]\to [0:0:1]$ as $y\to \infty$). Smoothness of $E$ implies $\Delta =-16(4a^3+27b^2)\ne 0$. The map is a closed embedding because $|3O|$ is very ample on the genus-1 curve. $\square$

The remaining advanced results — Mazur's torsion theorem, the modularity theorem, the BSD conjecture in the rank-$\le 1$ case, the main theorem of complex multiplication, and Tate's algorithm — are stated without proof here; primary references are listed in the Bibliography. Each is a self-contained body of work spanning multiple papers and book chapters.

Connections [Master]

• Riemann-Roch theorem for curves 04.04.01. Riemann-Roch on a genus-1 curve gives $\dim H^0(E,\mathcal{O}(D))=\mathrm{deg}D$ for $\mathrm{deg}D\ge 1$ (since $K_E=0$ on an elliptic curve), which is exactly the dimension count powering the Weierstrass embedding ${\varphi }_{|3O|}$ and the identification $E(k)\cong {\mathrm{Pic}}^0(E)$. Without Riemann-Roch the explicit Weierstrass model and the group structure on $E(k)$ have no quantitative engine.

• Hurwitz formula 04.04.02. A double cover $E\to {\mathbb{P}}^1$ via the Weierstrass $x$-coordinate is ramified at four points (the three roots of $x^3+ax+b$ together with the point at infinity), and Hurwitz reproduces $g_E=1$ from $2g_E-2=2\cdot (-2)+4=0$. Every elliptic curve is hyperelliptic in the elementary sense of admitting a degree-2 map to ${\mathbb{P}}^1$.

• Picard group 04.05.02. The map $E(k)\to {\mathrm{Pic}}^0(E)$, $P\mapsto [P-O]$, is a group isomorphism — every degree-zero divisor class on a genus-1 curve has a unique effective representative of the form $P-O$. The Jacobian of an elliptic curve coincides with the elliptic curve itself: $\mathrm{Jac}(E)=E$, the simplest case of the Jacobi inversion theorem 06.06.06.

• Jacobi inversion theorem 06.06.06. For a genus-1 curve $X$ over $\mathbb{C}$, the Abel-Jacobi map ${\mathrm{Sym}}^1(X)=X\to \mathrm{Jac}(X)$ is the identity on $\mathbb{C}/\Lambda$ (after fixing a reference point). The Jacobi inversion in genus 1 is the identification $X=\mathrm{Jac}(X)$, and the elliptic curve is the universal example for the higher-genus theory.

• Theta functions and modular forms. Over $\mathbb{C}$, an elliptic curve $E_{\tau }=\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ has Weierstrass coefficients $g_2(\tau )=60{\sum }_{(m,n)\ne 0}(m+n\tau {)}^{-4}$ and $g_3(\tau )=140{\sum }_{(m,n)\ne 0}(m+n\tau {)}^{-6}$, and the $j$-invariant $j(\tau )=1728g_2^3/(g_2^3-27g_3^2)$ is a modular function on $\mathfrak{H}$ generating the function field of $X(1)={\mathbb{P}}^1$. The Riemann theta function on the elliptic curve becomes the Jacobi theta functions ${\vartheta }_i(z;\tau )$, and its zeros are the 2-torsion points.

• Modular curves and modular forms. The modular curves $X_0(N),X_1(N)$ parametrise elliptic curves with level structure; the $L$-function of $E$ over $\mathbb{Q}$ is the $L$-function of a weight-2 cusp form on ${\Gamma }_0(N(E))$ by the modularity theorem. Heegner points on $X_0(N)$ give rational points on $E$ that detect the rank.

• Abelian varieties. Elliptic curves are the 1-dimensional abelian varieties; the Mordell-Weil theorem and the Hasse bound generalise to all abelian varieties over number fields and finite fields respectively. The Tate module $T_{\ell }E$ as a Galois representation, the étale cohomology $H_{\text{eˊt}}^1(E_{\bar{k}},{\mathbb{Q}}_{\ell })\cong V_{\ell }E$, and the Frobenius eigenvalue structure are all 1-dimensional cases of the higher-dimensional theory.

• Galois representations and the Langlands program. The 2-dimensional Galois representations ${\rho }_{E,\ell }:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$ associated to $E$ are among the simplest geometrically interesting Galois representations. Modularity identifies them with Galois representations attached to weight-2 modular forms — the foundational rank-2 case of the global Langlands correspondence between Galois representations and automorphic forms.

Historical & philosophical context [Master]

Karl Weierstrass developed the theory of elliptic functions and the Weierstrass equation in his Berlin lectures of the 1860s-1880s, summarised in the posthumous Vorlesungen über elliptische Funktionen edited by Schwarz (1893) ^{[Weierstrass]}. The Weierstrass $\wp$-function $\wp (z;\Lambda )$ — the unique even meromorphic function on $\mathbb{C}/\Lambda$ with double pole at $0$ and the leading-term normalisation $\wp (z)-z^{-2}\to 0$ as $z\to 0$ — satisfies the differential equation $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$, which is exactly the Weierstrass equation $y^2=4x^3-g_{2}x-g_3$ after the substitution $(x,y)=(\wp ,{\wp }^{\prime })$. The chord-and-tangent law on the cubic corresponds to the addition formula for $\wp$, $\wp (z+w)=-\wp (z)-\wp (w)+\frac{1}{4}({\wp }^{\prime }(z)-{\wp }^{\prime }(w){)}^2/(\wp (z)-\wp (w){)}^2$, which Weierstrass proved by a residue calculation.

Henri Poincaré in 1901 Sur les propriétés arithmétiques des courbes algébriques ^{[Poincaré 1901]} (J. de Math. (5) 7, 161-233) was the first to consider $E(\mathbb{Q})$ as an abelian group and to conjecture finite generation. Louis J. Mordell proved the conjecture for elliptic curves over $\mathbb{Q}$ in 1922 On the rational solutions of the indeterminate equations of the third and fourth degrees ^{[Mordell 1922]} (Proc. Camb. Phil. Soc. 21, 179-192), introducing the descent argument with the naive height as Step 2 and Selmer-style 2-descent as Step 1. André Weil generalised Mordell's theorem to all elliptic curves over number fields in his thesis L'arithmétique sur les courbes algébriques ^{[Weil 1929]} (Acta Math. 52, 281-315) and further to higher-dimensional abelian varieties — the Mordell-Weil theorem. The canonical-height construction is due to Néron 1965 and Tate 1965 (independently, via the Tate limit), and the modern formulation in terms of Néron's local heights at each place is in Lang's Fundamentals of Diophantine Geometry (1983).

Yutaka Taniyama in 1955 conjectured at the Tokyo-Nikko Symposium that every elliptic curve over $\mathbb{Q}$ is modular — every $L$-function $L(E,s)$ matches the $L$-function of a weight-2 cuspidal modular form. Goro Shimura refined the conjecture in print in his 1971 Introduction to the Arithmetic Theory of Automorphic Functions. Andrew Wiles proved the semistable case in 1995 ^{[Wiles 1995]} Modular elliptic curves and Fermat's Last Theorem (Ann. Math. 141, 443-551), with the gap in the original argument fixed by Wiles and Taylor Ring-theoretic properties of certain Hecke algebras (Ann. Math. 141, 553-572). Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended the proof to all elliptic curves over $\mathbb{Q}$ in On the modularity of elliptic curves over $\mathbb{Q}$ ^{[Breuil-Conrad-Diamond-Taylor 2001]} (J. Amer. Math. Soc. 14, 843-939), completing the modularity theorem.

Barry Mazur's Modular curves and the Eisenstein ideal ^{[Mazur 1977]} (Publ. Math. IHES 47, 33-186) determined the possible torsion subgroups of $E(\mathbb{Q})$ for all $E/\mathbb{Q}$, the first major application of the Eisenstein-ideal analysis of $J_0(N)$. John Tate's 1974 survey The arithmetic of elliptic curves ^{[Tate 1974]} (Invent. Math. 23, 179-206) consolidated the modern algebraic-geometric formulation, the conductor-discriminant theory, the Tate module $T_{\ell }E$ as a Galois representation, and the Tate-Shafarevich and Selmer groups, providing the framework within which Mazur, Wiles, and the BSD-conjecture program unfolded.

Bibliography [Master]

[object Promise]