Mathematics Lecture Notes

Definition4.16Elliptic curve

An elliptic curve over a field $k$ is a pair $(E, O)$ where $E$ is a smooth projective curve of genus $1$ over $k$ and $O \in E(k)$ is a specified $k$ -rational point (the origin).

Every elliptic curve admits a Weierstrass equation: an embedding $E \hookrightarrow \mathbb{P}^2$ as a cubic curve

$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6, \quad a_i \in k,$

with $O = [0:1:0]$ (the point at infinity on the $y$ -axis).

If $\operatorname{char}(k) \neq 2, 3$ , we can complete the square and cube to get the short Weierstrass form:

$y^2 = x^3 + Ax + B, \quad A, B \in k, \quad 4A^3 + 27B^2 \neq 0.$

The condition $4A^3 + 27B^2 \neq 0$ ensures smoothness (i.e., the discriminant $\Delta = -16(4A^3 + 27B^2) \neq 0$ ).

ExampleConcrete elliptic curves

$E: y^2 = x^3 - x$ over $\mathbb{Q}$ : Here $A = -1$ , $B = 0$ , $\Delta = 64 \neq 0$ . The curve has complex multiplication by $\mathbb{Z}[i]$ (the Gaussian integers), since $[i]: (x, y) \mapsto (-x, iy)$ is an endomorphism.
$E: y^2 = x^3 - 1$ over $\mathbb{Q}$ : Here $A = 0$ , $B = -1$ , $\Delta = -432 \neq 0$ . Has CM by $\mathbb{Z}[\omega]$ where $\omega = e^{2\pi i/3}$ , via $[\omega]: (x,y) \mapsto (\omega x, y)$ .
$E: y^2 = x^3 + x + 1$ over $\mathbb{F}_5$ : The points are $O, (0, \pm 1), (2, \pm 1), (3, \pm 1), (4, \pm 2)$ , so $|E(\mathbb{F}_5)| = 9$ .
$E: y^2 = x(x-1)(x-\lambda)$ for $\lambda \neq 0, 1$ : the Legendre family, a universal family of elliptic curves with full level-2 structure. The $j$ -invariant is $j = 256 \frac{(\lambda^2 - \lambda + 1)^3}{\lambda^2(\lambda - 1)^2}$ .

Definition4.17Group law on an elliptic curve

The set $E(k)$ of $k$ -rational points carries a natural abelian group structure with identity $O$ :

Geometric description (chord-tangent): Given $P, Q \in E(k)$ :

Draw the line $L$ through $P$ and $Q$ (tangent line if $P = Q$ ).
$L$ meets $E$ in a third point $R$ (by Bezout: a line meets a cubic in 3 points).
Define $P + Q$ as the reflection of $R$ over the $x$ -axis: if $R = (x_0, y_0)$ , then $P + Q = (x_0, -y_0 - a_1 x_0 - a_3)$ .

Divisor-theoretic description: Under the Abel–Jacobi isomorphism $E(k) \xrightarrow{\sim} \operatorname{Pic}^0(E)$ , $P \mapsto [P - O]$ :

$P + Q = R \iff [P - O] + [Q - O] = [R - O] \in \operatorname{Pic}^0(E).$

Equivalently, $P + Q + R' \sim 3O$ where $R'$ is the third intersection and $P + Q =$ reflection of $R'$ .

ExampleExplicit addition formulas

For $E: y^2 = x^3 + Ax + B$ in short Weierstrass form, $P = (x_1, y_1)$ , $Q = (x_2, y_2)$ :

Case $P \neq Q$ : The slope of $PQ$ is $\lambda = \frac{y_2 - y_1}{x_2 - x_1}$ . Then:

$x_3 = \lambda^2 - x_1 - x_2, \quad y_3 = \lambda(x_1 - x_3) - y_1.$

Case $P = Q$ (doubling): $\lambda = \frac{3x_1^2 + A}{2y_1}$ . Same formulas for $x_3, y_3$ .

Negation: $-P = (x_1, -y_1)$ .

Numerical example on $y^2 = x^3 - 5x + 8$ : Let $P = (1, 2)$ and $Q = (-1, -2\sqrt{3})$ ... Actually, let's use $P = (1, 2)$ (check: $4 = 1 - 5 + 8 = 4$ ✓). Doubling: $\lambda = \frac{3 - 5}{4} = -\frac{1}{2}$ , so $x_3 = \frac{1}{4} - 2 = -\frac{7}{4}$ , $y_3 = -\frac{1}{2}(1 + \frac{7}{4}) - 2 = -\frac{1}{2} \cdot \frac{11}{4} - 2 = -\frac{27}{8}$ .

Check: $\left(-\frac{27}{8}\right)^2 = \frac{729}{64}$ and $\left(-\frac{7}{4}\right)^3 - 5\left(-\frac{7}{4}\right) + 8 = -\frac{343}{64} + \frac{35}{4} + 8 = -\frac{343}{64} + \frac{560}{64} + \frac{512}{64} = \frac{729}{64}$ ✓

ExampleTorsion points

On $E: y^2 = x^3 - x$ over $\mathbb{Q}$ :

2-torsion ( $2P = O$ ): these are points with $y = 0$ , i.e., $(0,0)$ , $(1,0)$ , $(-1,0)$ . So $E[2](\mathbb{Q}) \cong (\mathbb{Z}/2)^2$ .
Over $\bar{\mathbb{Q}}$ : $E[n] \cong (\mathbb{Z}/n)^2$ for all $n \geq 1$ . This is a free $\mathbb{Z}/n$ -module of rank $2$ .
Mordell–Weil theorem: $E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}}$ . For this curve: $E(\mathbb{Q}) \cong (\mathbb{Z}/2)^2$ (rank $0$ , torsion only).
Mazur's theorem: For $E/\mathbb{Q}$ , the torsion subgroup $E(\mathbb{Q})_{\text{tors}}$ is isomorphic to $\mathbb{Z}/n$ for $1 \leq n \leq 10$ or $n = 12$ , or $\mathbb{Z}/2 \times \mathbb{Z}/2m$ for $1 \leq m \leq 4$ .

Definition4.18j-invariant

The $j$ -invariant of $E: y^2 = x^3 + Ax + B$ is

$j(E) = 1728 \cdot \frac{4A^3}{4A^3 + 27B^2} = -1728 \cdot \frac{(4A)^3}{\Delta}.$

Two elliptic curves over an algebraically closed field $\bar{k}$ are isomorphic if and only if they have the same $j$ -invariant.

The $j$ -invariant defines a map $j: \mathcal{M}_{1,1} \to \mathbb{A}^1$ which is an isomorphism of coarse moduli spaces. Every $j_0 \in \bar{k}$ is realized by some elliptic curve.

ExampleNotable j-invariants

$j = 0$ : $E: y^2 = x^3 + 1$ ( $A = 0$ ). Has automorphism group $\mathbb{Z}/6$ (generated by $(x,y) \mapsto (\omega x, -y)$ , $\omega^3 = 1$ ). CM by $\mathbb{Z}[\omega]$ , $\omega = e^{2\pi i/3}$ .
$j = 1728$ : $E: y^2 = x^3 + x$ ( $B = 0$ ). Has $\operatorname{Aut}(E) \cong \mathbb{Z}/4$ (generated by $(x,y) \mapsto (-x, iy)$ ). CM by $\mathbb{Z}[i]$ .
$j \neq 0, 1728$ : $\operatorname{Aut}(E) = \{\pm 1\} \cong \mathbb{Z}/2$ (the generic case).
$j = -2^{15} \cdot 3 \cdot 5^3$ : corresponds to the curve $X_0(11): y^2 + y = x^3 - x^2 - 10x - 20$ , the modular elliptic curve of conductor $11$ .
Over $\mathbb{F}_p$ : the supersingular elliptic curves form a finite set. For $p \geq 5$ , $E$ is supersingular iff $|E(\mathbb{F}_p)| = p + 1$ iff the Hasse invariant vanishes. There are approximately $p/12$ supersingular $j$ -invariants.

Definition4.19Complex uniformization

Over $\mathbb{C}$ , every elliptic curve is isomorphic to $\mathbb{C}/\Lambda$ for a lattice $\Lambda = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2 \subset \mathbb{C}$ (with $\omega_1/\omega_2 \notin \mathbb{R}$ ).

The isomorphism is given by the Weierstrass $\wp$ -function:

$\wp(z) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus 0} \left(\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2}\right)$

via $z \mapsto [\wp(z) : \wp'(z) : 1]$ . This satisfies $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$ where $g_2 = 60 G_4$ , $g_3 = 140 G_6$ are Eisenstein series.

The moduli space: $\tau = \omega_1/\omega_2 \in \mathfrak{H}$ (upper half-plane), and $\mathbb{C}/\Lambda_\tau \cong \mathbb{C}/\Lambda_{\tau'}$ iff $\tau' = \frac{a\tau + b}{c\tau + d}$ for some $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})$ . So the moduli space is $\mathfrak{H}/\operatorname{SL}_2(\mathbb{Z})$ .

ExampleLattices and elliptic curves

Square lattice $\Lambda = \mathbb{Z} + \mathbb{Z}i$ : gives $\tau = i$ , $j = 1728$ . The curve $y^2 = x^3 + x$ . The 4-fold symmetry $z \mapsto iz$ of the lattice corresponds to $(x,y) \mapsto (-x, iy)$ .
Hexagonal lattice $\Lambda = \mathbb{Z} + \mathbb{Z}\omega$ , $\omega = e^{2\pi i/3}$ : gives $\tau = \omega$ , $j = 0$ . The curve $y^2 = x^3 + 1$ . The 6-fold symmetry $z \mapsto \omega z$ gives $(x,y) \mapsto (\omega x, y)$ ... but the Weierstrass map mixes things up; the 6-fold symmetry descends to 3-fold on $x$ and sign flip on $y$ .
$\tau = 2i$ : a "tall" rectangular lattice. No extra symmetry, so $j \neq 0, 1728$ . The $j$ -invariant can be computed from $q = e^{2\pi i \cdot 2i} = e^{-4\pi}$ via the $q$ -expansion $j(\tau) = q^{-1} + 744 + 196884q + \cdots$ .
Degeneration: as $\tau \to i\infty$ (lattice becomes "infinitely tall"), $q \to 0$ and $j \to \infty$ . The curve degenerates to a nodal cubic $y^2 = x^3 + x^2$ (a "Neron 1-gon").

Definition4.20Isogeny

An isogeny $\phi: E_1 \to E_2$ is a morphism of elliptic curves that sends $O_1 \mapsto O_2$ . Equivalently, it is a surjective group homomorphism with finite kernel.

Key properties:

Every isogeny has finite kernel $\ker(\phi) \subset E_1(\bar{k})$ .
The degree of $\phi$ equals $|\ker(\phi)|$ (for separable isogenies; in general, $\deg \phi = |\ker \phi| \cdot p^i$ for some inseparable part).
The dual isogeny $\hat{\phi}: E_2 \to E_1$ satisfies $\hat{\phi} \circ \phi = [n]$ (multiplication by $n = \deg \phi$ ) on $E_1$ .
$\operatorname{End}(E) = \operatorname{Hom}(E, E)$ is the endomorphism ring: either $\mathbb{Z}$ , an order in an imaginary quadratic field (CM case), or a maximal order in a quaternion algebra (supersingular case in char $p$ ).

ExampleIsogenies

Multiplication by $n$ : $[n]: E \to E$ is an isogeny of degree $n^2$ , with $\ker([n]) = E[n] \cong (\mathbb{Z}/n)^2$ (if $(n, \operatorname{char} k) = 1$ ).
Frobenius: Over $\mathbb{F}_q$ , the Frobenius $\phi_q: (x,y) \mapsto (x^q, y^q)$ is an isogeny of degree $q$ . The trace $a_q = q + 1 - |E(\mathbb{F}_q)|$ satisfies $\phi_q^2 - a_q \phi_q + q = 0$ in $\operatorname{End}(E)$ .
2-isogenies from 2-torsion: If $P_0 = (e, 0) \in E[2]$ , there is a 2-isogeny $\phi: E \to E' = E/\langle P_0 \rangle$ given by Velu's formulas. For $E: y^2 = (x - e)(x^2 + ex + f)$ :

$\phi(x, y) = \left(x + \frac{f - e^2}{x - e},\; y \cdot \frac{(x-e)^2 - (f - e^2)}{(x-e)^2}\right).$

$\operatorname{End}(E)$ for CM curves: For $E: y^2 = x^3 - x$ , $\operatorname{End}(E) \cong \mathbb{Z}[i]$ with $i: (x,y) \mapsto (-x, iy)$ . The element $1 + i \in \mathbb{Z}[i]$ gives a $(1+i)$ -isogeny of degree $|1+i|^2 = 2$ .

TheoremHasse bound

For an elliptic curve $E$ over $\mathbb{F}_q$ :

$|E(\mathbb{F}_q)| = q + 1 - a_q, \quad |a_q| \leq 2\sqrt{q}.$

This is the "Riemann hypothesis for elliptic curves": the zeta function $Z(E/\mathbb{F}_q, T) = \frac{1 - a_q T + qT^2}{(1-T)(1-qT)}$ has its roots on the circle $|T| = q^{-1/2}$ .

ExamplePoint counting

$E: y^2 = x^3 + x + 1$ over $\mathbb{F}_5$ : We computed $|E(\mathbb{F}_5)| = 9$ . So $a_5 = 5 + 1 - 9 = -3$ . Check: $|-3| = 3 \leq 2\sqrt{5} \approx 4.47$ ✓.
Over $\mathbb{F}_7$ : for $y^2 = x^3 + 1$ , direct counting gives $|E(\mathbb{F}_7)| = 7$ (note: $7 \equiv 1 \pmod{3}$ ). So $a_7 = 1$ . The curve is ordinary.
Supersingular over $\mathbb{F}_p$ ( $p \geq 5$ ): $a_p = 0$ , so $|E(\mathbb{F}_p)| = p + 1$ . Example: $y^2 = x^3 + 1$ over $\mathbb{F}_5$ : points are $O$ , $(0, \pm 1)$ , $(2, \pm 2)$ , $(3, 0)$ . That's $6 = 5 + 1$ ✓, so this is supersingular over $\mathbb{F}_5$ .

ExampleThe Birch and Swinnerton-Dyer conjecture

The BSD conjecture (one of the Clay Millennium Problems) relates the arithmetic and analytic data of $E/\mathbb{Q}$ :

$\operatorname{ord}_{s=1} L(E, s) = \operatorname{rank} E(\mathbb{Q}),$

where $L(E, s) = \prod_p L_p(E, s)$ is the Hasse–Weil $L$ -function. The leading coefficient involves the regulator, the Tate–Shafarevich group $\text{III}(E/\mathbb{Q})$ , and other invariants.

Known cases:

If $\operatorname{ord}_{s=1} L(E,s) = 0$ : then $\operatorname{rank} E(\mathbb{Q}) = 0$ (Kolyvagin, using Gross–Zagier).
If $\operatorname{ord}_{s=1} L(E,s) = 1$ : then $\operatorname{rank} E(\mathbb{Q}) = 1$ (Gross–Zagier + Kolyvagin).
$\operatorname{ord}_{s=1} L(E,s) \geq 2$ : wide open.

Definition4.21Modular curves

The modular curve $Y(1) = \mathfrak{H}/\operatorname{SL}_2(\mathbb{Z})$ parametrizes isomorphism classes of elliptic curves over $\mathbb{C}$ via $\tau \mapsto \mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau)$ .

Its compactification $X(1) = Y(1) \cup \{\text{cusp}\} \cong \mathbb{P}^1$ via the $j$ -invariant.

With level structure:

$Y_0(N)$ parametrizes pairs $(E, C)$ where $C \subset E[N]$ is a cyclic subgroup of order $N$ .
$Y_1(N)$ parametrizes pairs $(E, P)$ where $P \in E[N]$ is a point of exact order $N$ .
$Y(N)$ parametrizes $(E, \alpha)$ where $\alpha: (\mathbb{Z}/N)^2 \xrightarrow{\sim} E[N]$ is a full level- $N$ structure.

ExampleModular curves and modularity

$X_0(11)$ : The first modular curve of genus $1$ . It is the elliptic curve $y^2 + y = x^3 - x^2 - 10x - 20$ of conductor $11$ .
$X_0(N)$ for small $N$ : genus $0$ for $N \leq 10$ and $N = 12, 13, 16, 18, 25$ . First genus- $1$ case: $N = 11$ . First genus- $2$ case: $N = 23$ .
Modularity theorem (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor): Every elliptic curve $E/\mathbb{Q}$ of conductor $N$ corresponds to a weight-2 newform $f \in S_2(\Gamma_0(N))$ , i.e., there exists a surjection $X_0(N) \to E$ .
Fermat's Last Theorem follows: Frey showed a solution $a^p + b^p = c^p$ gives a semistable curve $E: y^2 = x(x - a^p)(x + b^p)$ that cannot be modular by Ribet's theorem. Wiles proved all semistable curves are modular. Contradiction.

RemarkConnections and applications

Elliptic curves sit at a remarkable crossroads:

Number theory: Mordell–Weil, BSD, Fermat's Last Theorem, modular forms.
Algebraic geometry: simplest abelian varieties, starting point for curves of higher genus.
Complex geometry: $\mathbb{C}/\Lambda$ , moduli of lattices, modular forms.
Cryptography: ECDSA, pairing-based cryptography, isogeny-based post-quantum crypto (SIKE/SIDH).
Physics: mirror symmetry, string compactifications on elliptic fibrations.

Math Notes

Elliptic Curves

Definition and Weierstrass form

The group law

The $j$ -invariant and isomorphism classes

Elliptic curves over $\mathbb{C}$ : lattices and uniformization

Isogenies

Arithmetic of elliptic curves

Elliptic curves and moduli

Summary: the central role of elliptic curves