This course develops elliptic curves from the ground up, starting with plane cubic curves and ending with the group law that makes these curves central objects in number theory. The focus is on how a geometric picture leads to arithmetic structure: first by understanding cubic curves in the plane, then by moving to projective geometry and Weierstrass form, and finally by turning geometric constructions into explicit algebraic formulas and computations.

The early chapters build the geometric foundations, explaining smooth versus singular cubics, why the point at infinity is needed, and how chords and tangents define addition on the curve. From there, the course proves that this operation really does satisfy the group axioms, with associativity as the key conceptual hurdle. Later chapters explore the same curve from several viewpoints: as a real geometric object, as a computational tool over the rationals, and as a finite set over finite fields. The final chapters return to the geometry to explain flexes, collinearity, and the deeper reason the group law works, tying together the course's geometric intuition, algebraic formulas, and arithmetic applications.

# Introduction

These notes introduce elliptic curves through the geometry of plane cubic curves. The guiding question is how a geometric construction with lines and cubics produces an abelian group, and why that group can be computed over fields such as $\mathbb R$, $\mathbb Q$, and $\mathbb F_q$. The first part of the course builds the construction from projective geometry; the later parts translate it into explicit Weierstrass formulas and arithmetic examples.

The course begins before the phrase "elliptic curve" has a formal meaning. We first study cubic equations in two variables, discover why affine geometry loses intersection points, and then move to the projective plane where lines meet cubics with the correct count. This introduction records the main objects, the intended prerequisites, and the central theorem toward which the early lectures are directed.

## The Central Question

A line through two points on a cubic usually meets the cubic once more. The group law starts from the idea that this third intersection point should be converted into a sum, after choosing a distinguished point to act as the identity.

[explanation: Why Cubics Lead To Addition] For a conic, a line through two known points has already used up its two intersections. For a cubic, a line through two known points still has one intersection remaining, counted with multiplicity and after passing to projective space. This extra point is the geometric resource that makes an operation possible.

The operation is not merely a rule for drawing pictures. Once the cubic is nonsingular and a base point is chosen, the chord-and-tangent construction gives an abelian group on the set of points of the cubic over the field under consideration. The rest of the course is devoted to making each phrase in that sentence precise. [/explanation]

This viewpoint explains the order of the course. We cannot define the addition law responsibly until we know what points at infinity are, how tangencies count as repeated intersections, and why singular cubics must be excluded.

## Plane Cubics As The Starting Object

The first problem is to decide what kind of equation should be allowed. We want a class of curves large enough to contain examples such as $y^2=x^3-x$, but rigid enough that a line has a predictable intersection theory.

[definition: Affine Plane Cubic] Let $k$ be a field. An affine plane cubic over $k$ is a subset $C \subset k^2$ of the form \begin{align*} C = \{(x,y) \in k^2 : F(x,y)=0\}, \end{align*} where $F \in k[x,y]$ is a polynomial of total degree $3$. [/definition]

Affine cubics are easy to write down, but they are not the right final setting. Vertical lines may miss the expected third point, and two parallel lines never meet in the affine plane. These defects are not arithmetic phenomena; they come from using an incomplete geometric space.

[example: Missing Point At Infinity] Consider the affine cubic over $\mathbb R$ \begin{align*} y^2=x^3-x. \end{align*} Its projective closure is obtained by replacing $x$ by $X/Z$ and $y$ by $Y/Z$: \begin{align*} \left(\frac{Y}{Z}\right)^2=\left(\frac{X}{Z}\right)^3-\frac{X}{Z}. \end{align*} Multiplying both sides by $Z^3$ gives \begin{align*} Y^2Z=X^3-XZ^2. \end{align*} Fix $a \in \mathbb R$. The affine vertical line $x=a$ becomes the projective line $X=aZ$. Substituting $X=aZ$ into the projective cubic gives \begin{align*} Y^2Z=(aZ)^3-(aZ)Z^2. \end{align*} The right-hand side expands to \begin{align*} (aZ)^3-(aZ)Z^2=a^3Z^3-aZ^3=(a^3-a)Z^3. \end{align*} Hence the intersection points on this projective line satisfy \begin{align*} Y^2Z=(a^3-a)Z^3. \end{align*} Moving all terms to one side and factoring out $Z$ gives \begin{align*} Z\left(Y^2-(a^3-a)Z^2\right)=0. \end{align*} The factor $Z=0$ gives $X=aZ=0$, so the point has the form $[0:Y:0]$. Since projective coordinates cannot all be zero, $Y \ne 0$, and therefore \begin{align*} [0:Y:0]=[0:1:0]. \end{align*} If $a^3-a>0$ and $y^2=a^3-a$, the factor $Y^2-(a^3-a)Z^2=0$ gives, in the affine chart $Z=1$, the two points $[a:y:1]$ and $[a:-y:1]$, corresponding to $(a,y)$ and $(a,-y)$. Thus the affine plane sees only the two finite intersections, while the projective closure supplies the third point $[0:1:0]$, later used as the identity element for the Weierstrass group law. [/example]

This example is the first warning that the course cannot remain purely affine. The geometric operation becomes uniform only after replacing the affine plane by the projective plane.

## Projective Geometry Needed For Cubics

The next problem is to add the missing points without changing the equations beyond recognition. Projective coordinates achieve this by treating scalar multiples of nonzero triples as the same point.

[definition: Projective Plane] Let $k$ be a field. The projective plane over $k$ is \begin{align*} \mathbb P^2_k = (k^3 \setminus \{0\})/\sim, \end{align*} where $(X,Y,Z) \sim (\lambda X,\lambda Y,\lambda Z)$ for every $\lambda \in k^\times$. The equivalence class of $(X,Y,Z)$ is written $[X:Y:Z]$. [/definition]

The affine plane $k^2$ appears inside $\mathbb P^2_k$ as the chart $Z \ne 0$ by sending $(x,y)$ to $[x:y:1]$. The remaining line $Z=0$ is the line at infinity, where the missing intersections of affine geometry live.