Elliptic curves and projective geometry form a powerful duo in mathematics. They combine to create a rich framework for studying geometric objects with algebraic properties. This union allows for deeper insights into curve behavior, point counting, and group structures.

Projective space provides a natural setting for elliptic curves, offering a more complete view of their properties. By adding points at infinity and using homogeneous coordinates, we can explore the full range of curve behavior and develop efficient algorithms for cryptography and other applications.

Elliptic curves in projective space

• Projective space provides a natural setting for studying elliptic curves, allowing for a more unified treatment of their geometry and arithmetic
• Elliptic curves in projective space have additional structure and properties compared to their affine counterparts, such as the presence of points at infinity and a well-defined group law

Projective plane vs affine plane

• The projective plane extends the affine plane by adding points at infinity, providing a more complete geometric picture
• In the affine plane, parallel lines never intersect, while in the projective plane, parallel lines intersect at a point at infinity
• The projective plane can be constructed from the affine plane by adding a line at infinity, which consists of points representing the directions of lines in the affine plane

Homogeneous coordinates for projective space

• Homogeneous coordinates are used to represent points in projective space, allowing for a consistent treatment of points at infinity
• A point in projective space is represented by a triple $[X:Y:Z]$, where $X$, $Y$, and $Z$ are not all zero, and $[X:Y:Z]$ is considered equivalent to $[λX:λY:λZ]$ for any non-zero scalar $λ$
• The affine plane can be embedded in the projective plane by setting $Z=1$, so that $(x,y)$ in the affine plane corresponds to $[x:y:1]$ in the projective plane

Points at infinity on elliptic curves

• Elliptic curves in projective space have a unique point at infinity, denoted by $O$ or $[0:1:0]$, which serves as the identity element for the group law
• The point at infinity arises from the intersection of the elliptic curve with the line at infinity in the projective plane
• Adding the point at infinity to an elliptic curve completes its group structure and allows for a more unified treatment of the curve's geometry and arithmetic

Group law in projective coordinates

• The group law on an elliptic curve in projective space is defined geometrically using the chord-and-tangent process, similar to the affine case
• To add two points $P$ and $Q$ on an elliptic curve in projective space, draw a line through $P$ and $Q$ (or a tangent line if $P=Q$), and find the third point of intersection $R$ with the curve. The sum $P+Q$ is defined as the point obtained by reflecting $R$ across the $x$-axis
• The point at infinity serves as the identity element for the group law, so that $P+O=P$ for any point $P$ on the curve

Projective equations of elliptic curves

• Elliptic curves in projective space can be defined by homogeneous equations of the form $Y^2Z=X^3+aXZ^2+bZ^3$, where $a$ and $b$ are constants
• The projective equation is obtained from the affine Weierstrass equation $y^2=x^3+ax+b$ by homogenizing the variables: $x=X/Z$ and $y=Y/Z$
• The projective equation allows for the inclusion of the point at infinity, which corresponds to $[0:1:0]$ and satisfies the equation

Intersection of curves in projective space

• The intersection of curves in projective space is a fundamental concept in algebraic geometry, with important applications to the study of elliptic curves
• Intersecting curves in projective space can be analyzed using tools such as Bézout's theorem and intersection multiplicity, which provide insights into the number and nature of intersection points

Bézout's theorem

• Bézout's theorem states that two projective plane curves of degrees $m$ and $n$ intersect in exactly $mn$ points, counted with multiplicity, unless the curves have a common component
• The theorem provides an upper bound on the number of intersection points between two curves in projective space
• Bézout's theorem is a powerful tool for studying the intersection of elliptic curves with other curves, such as lines and conics

Intersection multiplicity

• Intersection multiplicity is a measure of the complexity of the intersection between two curves at a given point
• If two curves intersect at a point with multiplicity $k$, then the curves share $k$ coincident points at that intersection
• The concept of intersection multiplicity is crucial for understanding the behavior of curves at singular points and for computing the total number of intersections between curves

Tangent lines and multiplicity

• A tangent line to a curve at a point is a line that intersects the curve with multiplicity at least 2 at that point
• The multiplicity of the intersection between a curve and its tangent line at a point is related to the singularity of the curve at that point
• For elliptic curves, the tangent line at a point $P$ intersects the curve at exactly one other point, which is used to define the group law on the curve

Singular points on curves

• A singular point on a curve is a point where the curve fails to be smooth, i.e., where the tangent line is not well-defined or has higher multiplicity
• Singular points on elliptic curves are classified into two types: nodes (ordinary double points) and cusps
• The presence and nature of singular points on an elliptic curve provide important information about its geometry and arithmetic, such as its conductor and reduction type

Elliptic curves as cubic curves

• Elliptic curves are a special class of cubic curves in projective space, characterized by their smooth structure and the presence of a distinguished point (the point at infinity)
• The study of elliptic curves as cubic curves allows for a deeper understanding of their geometric properties and their connections to other areas of mathematics, such as algebraic geometry and number theory

Weierstrass equations in projective space

• Elliptic curves in projective space can be described by homogeneous Weierstrass equations of the form $Y^2Z=X^3+aXZ^2+bZ^3$, where $a$ and $b$ are constants
• The projective Weierstrass equation is obtained by homogenizing the affine Weierstrass equation $y^2=x^3+ax+b$, allowing for the inclusion of the point at infinity
• The coefficients $a$ and $b$ in the Weierstrass equation determine the geometric and arithmetic properties of the elliptic curve, such as its discriminant and j-invariant

Projective transformations of Weierstrass form

• Projective transformations can be applied to the Weierstrass equation of an elliptic curve to simplify its form or to study its isomorphism class
• A projective transformation is a change of variables of the form $X=uX'+rZ'$, $Y=vY'+sZ'$, $Z=wZ'$, where $u,v,w,r,s$ are constants with $uvw≠0$
• Projective transformations preserve the geometric structure of the elliptic curve, including its group law and the number of rational points

Discriminant and j-invariant

• The discriminant $Δ$ of an elliptic curve in Weierstrass form is a polynomial expression in the coefficients $a$ and $b$, given by $Δ=-16(4a^3+27b^2)$
• The discriminant determines the singularity of the elliptic curve: the curve is smooth if and only if $Δ≠0$, and it has a node or a cusp if $Δ=0$
• The j-invariant of an elliptic curve is a rational expression in the coefficients $a$ and $b$, given by $j=1728\frac{4a^3}{4a^3+27b^2}$ (when $Δ≠0$), which characterizes the isomorphism class of the curve

Isomorphism classes of elliptic curves

• Two elliptic curves are said to be isomorphic if there exists a projective transformation that maps one curve onto the other, preserving the group structure
• Isomorphic elliptic curves share the same j-invariant, and over an algebraically closed field, the j-invariant completely determines the isomorphism class of the curve
• Understanding isomorphism classes of elliptic curves is important for classification purposes and for studying their moduli spaces

Rational points on elliptic curves

• The study of rational points on elliptic curves is a central topic in arithmetic geometry, with connections to diophantine equations and cryptography
• Rational points are points on an elliptic curve whose coordinates are rational numbers, forming a finitely generated abelian group under the curve's group law

Mordell-Weil theorem

• The Mordell-Weil theorem states that the group of rational points on an elliptic curve over a number field is finitely generated, i.e., it is isomorphic to $\mathbb{Z}^r \oplus T$, where $r$ is the rank of the curve and $T$ is the torsion subgroup
• The rank of an elliptic curve measures the "size" of its rational point group and is a key invariant in the study of elliptic curves
• Computing the rank of an elliptic curve is a difficult problem, related to the Birch and Swinnerton-Dyer conjecture, one of the Millennium Prize Problems

Torsion subgroup of rational points

• The torsion subgroup of an elliptic curve over a field $K$ consists of the points of finite order under the group law, i.e., points $P$ such that $nP=O$ for some positive integer $n$
• Over $\mathbb{Q}$, the torsion subgroup of an elliptic curve is completely classified by Mazur's theorem, which states that it is isomorphic to one of 15 possible groups
• Studying the torsion subgroup of an elliptic curve provides insights into its arithmetic structure and can be used to construct rational points of infinite order

Height functions and descent

• Height functions are tools used to measure the "size" or "complexity" of rational points on an elliptic curve, providing a way to study the arithmetic properties of the curve
• The canonical height is a quadratic form on the group of rational points, satisfying certain properties such as the parallelogram law, and is used to define the regulator of the curve
• Descent is a technique used to compute the rank of an elliptic curve by studying its rational points modulo various primes and using local-global principles to obtain information about the global structure of the rational point group

Weak Mordell-Weil theorem

• The weak Mordell-Weil theorem states that the group of rational points on an elliptic curve over a number field is a finitely generated abelian group
• The weak Mordell-Weil theorem is a key step in the proof of the full Mordell-Weil theorem, providing a finiteness result for the rational point group
• The proof of the weak Mordell-Weil theorem relies on techniques from diophantine geometry, such as height functions and the Néron-Tate pairing, to bound the size of the rational point group

Elliptic curves over finite fields

• The study of elliptic curves over finite fields has important applications in cryptography and coding theory, as well as connections to other areas of mathematics, such as number theory and algebraic geometry
• Over a finite field $\mathbb{F}_q$, an elliptic curve is a smooth projective curve of genus 1 with a distinguished point, and its rational points form a finite abelian group

Projective coordinates over finite fields

• Projective coordinates are particularly useful for studying elliptic curves over finite fields, as they allow for efficient computation and avoid the need for division operations
• In projective coordinates, an elliptic curve over $\mathbb{F}_q$ can be described by a homogeneous Weierstrass equation of the form $Y^2Z=X^3+aXZ^2+bZ^3$, where $a,b \in \mathbb{F}_q$
• Point addition and doubling formulas in projective coordinates involve only field operations and can be optimized for efficient implementation in cryptographic protocols

Group structure of elliptic curves over finite fields

• The group of $\mathbb{F}_q$-rational points on an elliptic curve, denoted by $E(\mathbb{F}_q)$, is a finite abelian group with a well-defined group law
• The structure of $E(\mathbb{F}_q)$ is determined by the Hasse-Weil bound, which states that the order of the group satisfies $|E(\mathbb{F}_q)| = q+1-t$, where $|t| \leq 2\sqrt{q}$
• The group structure of $E(\mathbb{F}_q)$ can be either cyclic or the product of two cyclic groups, depending on the number of points and the presence of torsion points

Hasse's theorem on point counts

• Hasse's theorem provides a precise bound on the number of $\mathbb{F}_q$-rational points on an elliptic curve, stating that $|E(\mathbb{F}_q)| = q+1-t$, where $|t| \leq 2\sqrt{q}$
• The value of $t$ in Hasse's theorem is called the trace of Frobenius and is related to the zeta function of the elliptic curve and its L-function
• Hasse's theorem has important consequences for the security of elliptic curve cryptography, as it guarantees that the group of rational points has a sufficiently large size and a well-distributed structure

Supersingular vs ordinary elliptic curves

• Elliptic curves over finite fields can be classified into two types: supersingular and ordinary, based on their endomorphism ring structure and their behavior under the Frobenius endomorphism
• An elliptic curve $E$ over $\mathbb{F}_q$ is called supersingular if the trace of Frobenius satisfies $t \equiv 0 \pmod{p}$, where $q=p^n$ for some prime $p$ and integer $n$; otherwise, it is called ordinary
• Supersingular elliptic curves have special properties, such as a larger automorphism group and a simpler group structure, and have applications in cryptography and coding theory, while ordinary curves are more commonly used in general elliptic curve cryptography