*2.10. Affine and Projective Curves

In this section, we introduce some elementary concepts from algebraic geometry, which facilitate the treatment of elliptic and hyperelliptic curves in the next two sections. We concentrate only on plane curves, because these are the only curves we need in this book. Throughout this section, K denotes a field (finite or infinite) and the algebraic closure of K.

2.10.1. Plane Curves

The solutions of a polynomial equation f(X, Y) = 0 is one of the central objects of study in algebraic geometry. For example, we know that in the equation X2 + Y2 – 1 = 0 represents a circle with origin at (0, 0) and with radius 1. When we pass to an arbitrary field, it is often not possible to visualize such plots, but it still makes sense to talk about the set of solutions of such an equation. For example, the solutions of the above circle in are the four discrete points (0, 1), (0, 2), (1, 0) and (2, 0). (This solution set does not really look round.)

One can generalize this study by considering polynomials in n indeterminates and by investigating the simultaneous solutions of m polynomials. We, however, do not intend to be so general here and concentrate only on curves defined by a single polynomial equation in two indeterminates.

Definition 2.61.

For , the n-dimensional affine space over K is defined to be the set consisting of all n-tuples (x1, . . . , xn) with each . For n = 2, the affine space is also called the affine plane over K. For a point , the elements are called the affine coordinates of P. The affine space over the closure is often abbreviated as , when the field K is understood from the context.

is an n-dimensional vector space over K. For example, the affine plane can be identified with the conventional X-Y plane.

Definition 2.62.

An affine plane (algebraic) curve C over K is defined by a polynomial and is written as C : f(X, Y) = 0. The set C(K) of K-rational points on an affine plane curve C : f(X, Y) = 0 is the set of all points satisfying f(x, y) = 0.

K-rational points on a plane curve are precisely the solutions of the defining polynomial equation. Standard examples of affine plane curves include the straight lines given by aX + bY + c = 0, a, , not both 0, and the conic sections (circles, ellipses, parabolas and hyperbolas) given by aX2 + bXY + cY2 + dX + eY + f = 0, a, b, c, d, e, with at least one of a, b, c non-zero. For , the set of K-rational points can be drawn as a graph of the polynomial equation, whereas for an arbitrary field K (in particular, for finite fields) such drawings make little or no sense. However, it is often helpful to visualize curves as curves over (also called real curves) and then generalize the situation to an arbitrary field K.

The number ∞ is not treated as a real number (or integer or natural number). But it is often helpful to extend the definition of by including two points that are infinitely far away from the origin, one in each direction. This gives us the so-called extended real line . An immediate advantage of such a completion of is that every Cauchy sequence converges in . But for studying the roots of polynomial equations it is helpful to add only a single point at infinity to in order to get what is called the projective line over . Similarly, if we start with the affine plane and add a point at infinity for each slope of straight lines Y = aX + b and one more for the lines X = c, we get the so-called projective plane over . We also call the line passing through all the points at infinity in to be the line at infinity. An immediate benefit of passing from to is that in any two distinct lines (parallel or not in ) meet at exactly one point and through any two distinct points of passes a unique line.

Now it is time to replace by an arbitrary field K and rephrase our definitions in such a way that it continues to make sense to talk about points and line at infinity, even when K itself contains only finitely many points.

Definition 2.63.

Let . Define the relation ~ on the ‘punctured’ n + 1-dimensional affine space over K by (x0, . . . , xn) ~ (y0, . . . , yn) if and only if there exists a such that yi = λxi for all i = 0, . . . , n. It is easy to see that ~ is an equivalence relation on . The set of all equivalence classes of ~ is called the n-dimensional projective space over K. In particular, is called the projective plane over K. A point is the equivalence class of a point . The elements constitute a set of homogeneous coordinates for P.

It is evident that can be identified with the set of all 1-dimensional vector subspaces (that is lines) of the affine space . To argue that this formal definition tallies with the intuitive notion for n = 2 and , consider the affine 3-space referred to by the coordinates X, Y, Z. Look at the family of planes , parallel to the X-Y plane. (ε0 is the X-Y plane itself.) First take a non-zero value of λ, say λ = 1. Every line in passing through the origin and not parallel to the X-Y plane meets ε1 exactly at one point. Conversely, a unique line passes through each point on ε1 and the origin. In this way, we associate points of with points on ε1. These are all the finite points of . On the other hand, the lines passing through the origin and lying in the X-Y plane (ε0 : Z = 0) do not meet ε1 and correspond to the points at infinity of .

In the last paragraph, we obtained the canonical embedding of the affine plane in by setting Z = 1. By definition, is symmetric in X, Y and Z. This means that we can as well set X = 1 or Y = 1 and see that there are other embeddings of in . This observation often proves to be useful (for example, see Definition 2.66).

Now that we have passed from the affine plane to the projective plane, we should be able to carry (affine) plane curves to the projective plane. For this, we need some definitions.

Definition 2.64.

Let R denote the polynomial ring K[X0, X1, . . . , Xn] over a field K. A monomial of R is an element of R of the form , αi ≥ 0. A term in R is a monomial multiplied by an element . Any polynomial is a sum of finitely many nonzero terms. The degree of a monomial (or a term ) is defined as α0 + α1 + · · · + αn. The degree of a non-zero polynomial , denoted deg f, is defined to be the maximum of the degrees of its non-zero terms. The degree of the zero polynomial is taken to be –∞. A non-zero polynomial is said to be homogeneous of degree d ≥ 0, if all of its non-zero terms have degree d. The zero polynomial is said to be homogeneous of any degree.

Let C : f(X, Y) = 0 be an affine plane curve over a field K defined by a non-zero polynomial and d := deg f. Then f(h)(X, Y, Z) := Zdf(X/Z, Y/Z) is a homogeneous polynomial of degree d in the polynomial ring K[X, Y, Z]. The polynomial f(h) is called the homogenization of f. Putting Z = 1 in f(h)(X, Y, Z) gives back the original polynomial f(X, Y), that is, f(h)(X, Y, 1) = f(X, Y). Therefore, f is called the dehomogenization of the homogeneous polynomial f(h). The homogenization (and dehomogenization) of the zero polynomial is taken to be the zero polynomial.

Take and . By definition, [x, y, z] = [λx, λy, λz]. Since f(h)(λx, λy, λz) = λdf(h)(x, y, z) = 0 if and only if f(h)(x, y, z) = 0, it makes sense to talk about the zeros of the homogeneous polynomial f(h) in the projective plane . This motivates us to define projective plane curves:

Definition 2.65.

A projective plane curve C over K is defined by a homogeneous polynomial and is written as C : h(X, Y, Z) = 0. The set C(K) of K -rational points on a projective plane curve C : h(X, Y, Z) = 0 is the set of all points such that h(x, y, z) = 0.

Let C : f(X, Y) = 0 be an affine plane curve. The projective plane curve defined by f(h)(X, Y, Z) is by an abuse of notation denoted also by C. The zeros of the affine curve C : f(X, Y) = 0 in are in one-to-one correspondence with the finite zeros of C : f(h)(X, Y, Z) = 0 in (that is, zeros with Z = 1). The projective curve contains some more point(s), namely those at infinity, that can be obtained by putting Z = 0 in f(h)(X, Y, Z). Passage from the affine plane to the projective plane is just that: a systematic inclusion of the points at infinity.

It is often customary to write an affine plane curve as C : f(X, Y) = g(X, Y) and a projective plane curve as C : f(h)(X, Y, Z) = g(h)(X, Y, Z) with f(h) and g(h) of the same degree. The former is the same as the curve C : f – g = 0, and the latter the same as C : f(h) – g(h) = 0.

A homogeneous polynomial can be viewed as the homogenization of any of the polynomials

fZ(X, Y) = f(X, Y, 1), fY (X, Z) = f(X, 1, Z) and fX(Y, Z) = f(1, Y, Z).

Consider a point P = [a, b, c] on the projective curve C : f(X, Y, Z) = 0. Since a, b and c are not all 0, P is a finite point on at least one of fX, fY and fZ.

2.10.2. Polynomial and Rational Functions on Plane Curves

Throughout the rest of Section 2.10 we make the following assumption:

Assumption 2.1.

K is an algebraically closed field, that is, .

Although many of the results we state now are valid for fields that are not algebraically closed, it is convenient to make this assumption in order to avoid unnecessary complications.

Let C : f(X, Y) = 0 be a curve defined over K. Henceforth we assume that the polynomial f(X, Y) is irreducible over K. Though we write the affine equation for the curve for notational simplicity, we usually work with the set C(K) of the K-rational points on the corresponding projective curve. We refer to the solutions of C in the affine plane as the finite points on the curve.

Definition 2.66.

Let P = [a, b, c] be a point on a curve C defined over K. We call P a smooth or regular or non-singular point of C, if P satisfies the following conditions.

  1. If P is a finite point (that is, if c ≠ 0), then P is called a smooth point on C, if the partial derivatives ∂f/∂X and ∂f/∂Y do not vanish simultaneously at (a/c, b/c).

  2. If P is a point at infinity (that is, if c = 0), then we must have a ≠ 0 or b ≠ 0. Assume a ≠ 0. (The other case can be treated similarly.) Consider the polynomial . P is a finite point on the curve D : g(Y, Z) = 0. P is called a smooth point on C, if (b/a, 0) is a smooth point on D, that is, if ∂g/∂Y and ∂g/∂Z do not vanish simultaneously at (b/a, 0).

A non-smooth point on C is also called non-regular or singular. C is called smooth or regular or non-singular, if all points (finite and infinite) on C are smooth.

Now we define polynomial functions on C. For a moment, we concentrate on the affine curve, that is, only the finite points on C. Let g, with (that is, f|(g – h)). Since for any point P on C we have f(P) = 0, it follows that g(P) = h(P). This motivates us to define the following.

Definition 2.67.

The ring K[X, Y]/〈f〉 is called the affine coordinate ring of C and is denoted by K[C]. Elements of K[C] are called polynomial functions on C. If we denote by x and y the residue classes of X and Y respectively in K[C], then a polynomial function on C is given by a polynomial .[11] By our assumption, f is an irreducible polynomial; so 〈f〉 is a prime ideal of K[X, Y], that is, the coordinate ring K[C] is an integral domain.

[11] Recall from Section 2.7 that K[x, y] is the K-algebra generated by x and y. It is not a polynomial algebra (in general).

The quotient field (Exercise 2.34) of K[C] is called the function field of C and is denoted by K(C). An element of K(C) is of the form g(x, y)/h(x, y) with g(x, y), , h(x, y) ≠ 0 (that is, h(X, Y) ∉ 〈f〉), and is called a rational function on C.

By definition, two rational functions are equal if and only if g1(x, y)h2(x, y) – g2(x, y)h1(x, y) = 0 in K[C] or, equivalently, if and only if . We define addition and multiplication of rational functions by the usual rules (Exercise 2.34).

Definition 2.68.

Let P = (a, b) be a finite point on the curve C. Given a polynomial function , the value of g at P is defined to be . If is a rational function, then r is said to be defined at P, if r has a representation r = g/h, g, , with h(P) ≠ 0. In that case, we define the value of r at P to be . If r is not defined at P, it is customary to write r(P) = ∞.

By definition, K[C] and K(C) are collections of equivalence classes. However, the value of a polynomial or a rational function on C is independent of the representatives of the equivalence classes and is, therefore, a well-defined concept.

The above definitions can be extended to the corresponding projective curve C : f(h)(X, Y, Z) = 0. By Exercise 2.96(e), the polynomial f(h) is irreducible, since we assumed f to be so.

Definition 2.69.

The function field (denoted again by K(C)) of the projective curve C is the set of quotients (called rational functions) of the form g(X, Y, Z)/h(X, Y, Z), where g, are homogeneous of the same degree and h ∉ 〈f(h)〉. Two rational functions g1/h1 and g2/h2 are equal if and only if .

A rational function is said to be defined at a point P = [a, b, c] on C, if r has a representation g/h with h(a, b, c) ≠ 0. In that case, we define r(P) := g(a, b, c)/h(a, b, c). Since g and h are homogeneous and of the same degree, the value r(P) is independent of the choice of the projective coordinates of P (Exercise 2.95). If r is not defined at P, we write r(P) = ∞.

One can define polynomial functions on a projective curve (as we did for affine curves), but it makes no sense to talk about the value of such a polynomial function at a point P on the curve, because this value depends on the choice of the homogeneous coordinates of P (Exercise 2.95). This problem is eliminated for a rational function g/h by assuming g and h to be of the same degree.

Definition 2.70.

Let C be a projective plane curve, r be a non-zero rational function and P a point on C. P is called a zero of r if r(P) = 0, and a pole of r if r(P) = ∞.

Now we define the multiplicities of zeros and poles of a rational function or, more generally, the order of any point on a projective plane curve. This is based on the following result, the proof of which is long and difficult, and is omitted.

Theorem 2.41.

Let C be a projective plane curve defined by an irreducible polynomial over K and P a smooth point on C. Then there exists a rational function (depending on P) with the following properties:

  1. uP (P) = 0.

  2. For any non-zero rational function , there exist an integer d and a rational function having neither a zero nor a pole at P such that . The integer d does not depend on the choice of uP.

Definition 2.71.

The function uP of the last theorem is called a uniformizing variable or a uniformizing parameter or simply a uniformizer of C at P. For any non-zero rational function , the integer d with is called the order of r at P and is denoted by ordP (r).

The connection of poles and zeros with orders is established by the following theorem which we again avoid to prove.

Theorem 2.42.

P is neither a pole nor a zero of r if and only if ordP(r) = 0. P is a zero of r if and only if ordP(r) > 0. P is a pole of r if and only if ordP(r) < 0.

If P is a zero (resp. a pole) of r, the integer ordP(r) (resp. – ordP(r)) is called the multiplicity of the zero (resp. pole) P.

Theorem 2.43.

Let r be a rational function on the projective plane curve C defined over K. Then r has finitely many poles and zeros. Furthermore, .

This is one of the theorems that demand K to be algebraically closed. More explicitly, if K is not algebraically closed, any rational function continues to have only finitely many zeros and poles, but the sum of the orders of r at these points is not necessarily equal to 0. Also note that this sum, if taken over only the finite points of C, need not be 0, even when K is algebraically closed.

2.10.3. Maps Between Plane Curves

Now that we know how to define and evaluate rational functions on a curve, we are in a position to define rational maps between two curves. Let C1 : f1(X, Y, Z) = 0 and C2 : f2(X, Y, Z) = 0 be two projective plane curves defined over K by irreducible homogeneous polynomials f1, .

Definition 2.72.

A rational map (defined over K) is given by rational functions , , in K(C1) such that for each at which all of , and are defined, the point . One often uses the notation .

This, however, is not the complete story. A more precise characterization of a rational map is as follows:

A rational map is said to be defined at , if there exists a rational function (depending on P) such that , and are all defined at P, , and are not all zero and . A rational map which is defined at every point of C1(K) is called a morphism.

The curves C1 and C2 are said to be isomorphic (denoted C1 ≅ C2), if there exist morphisms and ψ : C2 → C1 such that and are identity maps on C1(K) and C2(K) respectively.

Isomorphism is an equivalence relation on the set of all projective plane curves defined over K. Since two isomorphic curves share many common algebraic and geometric properties, it is of interest in algebraic geometry to study the equivalence classes (rather than the individual curves). If C1 ≅ C2 and C2 has a simpler representation than C1, then studying the properties of C2 makes our job simpler and at the same time reveals all the common properties of C1. (See Section 2.11 for an example.)

**2.10.4. Divisors on Plane Curves

Let a be a symbol and n a positive integer. We represent by na the formal sum a+···+a (n times). We also define 0a := 0 and –na := n(–a), where the symbol –a satisfies a + (–a) = (–a) + a = 0. For n1, , we define n1a + n2a := (n1 + n2)a. The set under these definitions becomes an Abelian group. If we are given two symbols a, b we can analogously define formal sums na + mb, n, , and the sum of formal sums as (n1a + m1b) + (n2a + m2b) := (n1 + n2)a + (m1 + m2)b. With these definitions the set becomes an Abelian group. These constructions can be generalized as follows:

Definition 2.73.

Given a set (not necessarily finite) of symbols ai, , the set of formal sums of the form , where ni = 0 except for finitely many , is an Abelian group with the addition formula . This group is called the free Abelian group generated by ai, .

Now let ai be the K-rational points on a projective plane curve C defined over K. For notational convenience, we represent by [P] the symbol corresponding to the point P on C. This removes confusions in connection with elliptic curves C (See Section 2.11) for which we intend to make a distinction between P + Q and [P] + [Q] for two points P, . The former sum is again a point on C, whereas the latter is never (the symbol corresponding to) a point on C.

Definition 2.74.

A formal sum , , where nP = 0 except for finitely many , is called a divisor on C. The free Abelian group generated by the symbols [P] for all the points is called the group of divisors of C and is denoted by DivK(C) or simply by Div(C), when K is implicit in the context.

Let be a divisor. The support of D is defined to be the set and is denoted by Supp D.

The degree of D is defined as the integer and is denoted as deg D. The subset of Div(C) is clearly a subgroup of Div(C). We denote this subgroup by Div0(C).

Now we define divisors of rational functions on C. Henceforth we assume that C is smooth (that is, smooth at all K-rational points on C).

Definition 2.75.

The divisor of a rational function is defined to be the formal sum , where ordP(r) is the order of r at P (Definition 2.71). By Theorem 2.43 .

A divisor is called principal, if D = Div(r) for some rational function . We have Div(rr′) = Div(r) + Div(r′) for any rational functions r, . It follows that the set of all principal divisors on C is a subgroup of Div(C) (and of Div0(C) as well). We denote this subgroup by PrinK(C) or simply by Prin(C). The quotient group Div(C)/Prin(C) is called the divisor class group or the Picard group of C and is denoted by PicK(C) or in short by Pic(C). On the other hand, the quotient Div0(C)/Prin(C) is denoted by or Pic0(C) and is called the Jacobian of C. Instead of Pic0(C) we use the notation or .

Though the Jacobian is defined for an arbitrary smooth curve C (defined by an irreducible polynomial), it is a special class of curves called hyperelliptic curves for which it is particularly easy to represent and do arithmetic in the group . This gives us yet another family of groups on which cryptographic protocols can be built.

If K is not algebraically closed, we need not have for a rational function . This means that in that case the group cannot be defined in the above manner. However, since C is also a curve defined over , we can define as above and call a particular subgroup of as the Jacobian of C over K. We defer this discussion until Section 2.12.

Exercise Set 2.10

In this exercise set, we do not assume (unless otherwise stated) that K is necessarily algebraically closed.

2.95

  1. For homogeneous polynomials f1, of respective degrees d1 and d2, prove the following assertions:

    1. If d1 = d2, then f1 ± f2 are homogeneous polynomials of degree d1.

    2. The polynomial f1f2 is homogeneous of degree d1 + d2. Conversely, if f1f2 is homogeneous, then f1 and f2 are also homogeneous.

  2. A polynomial is homogeneous of degree d if and only if it satisfies f(λX1, . . ., λXn) = λdf(X1, . . ., Xn) for every .

2.96In this exercise, we generalize the notion of homogenization and dehomogenization of polynomials. Let K[X1, . . . , Xn] denote the polynomial ring in n indeterminates. Introducing another indeterminate X0, we define the homogenization of a polynomial as

Prove the following assertions.

  1. f(h) is an element of K[X0, X1, . . . , Xn] and is homogeneous of degree d.

  2. f(h)(1, X1, . . . , Xn) = f(X1, . . . , Xn).

  3. If deg f = d ≥ 0 and fd is the sum of all non-zero terms of degree d in f, then we have f(h)(0, X1, . . . , Xn) = fd(X1, . . . , Xn).

  4. For f, , (fg)(h) = f(h)g(h). Moreover, if g|f, then g(h)|f(h) and (f/g)(h) = f(h)/g(h). Under what condition(s) is (f + g)(h) = f(h) + g(h)?

  5. f is irreducible if and only if f(h) is irreducible.

2.97Let C : f(X, Y) = 0 be an affine plane curve defined by a non-zero polynomial and C : f(h)(X, Y, Z) = 0 the corresponding projective plane curve. Let d := deg f = deg f(h) and fd the sum of non-zero terms of f of degree d. Show that:

  1. f(h)(X, Y, 1) = f(X, Y) and f(h)(X, Y, 0) = fd(X, Y).

  2. is a K-rational point of the affine curve if and only if is a K-rational point of the projective curve. More generally, let . The point is a K-rational solution of f if and only if [x, y, λ] is a K-rational solution of f(h).

  3. The solutions of f at infinity are obtained by solving f(h)(X, Y, 0) = fd(X, Y) = 0. Conclude that the curve C can have at most d points at infinity.

  4. For a, , each of the curves Y – aX = b and X – aY = b (straight lines), and Y – X2 = 0 and X – Y2 = 0 (parabolas) contains only one point at infinity. The hyperbola XY – 1 = 0 contains two points at infinity. How many points at infinity does the hyperbola X2 – Y2 – 1 = 0 contain? The circle X2 + Y2 – 1 = 0?

  5. For a1, a2, a3, a4, , the elliptic curve Y2 + a1XY + a3Y = X3 + a2X2 + a4X + a6 contains only one point at infinity.

  6. Let and u(X), with deg u ≤ g, deg v = 2g + 1 and v monic. Show that the hyperelliptic curve Y2 + u(X)Y = v(X) has only one point at infinity.

2.98Show that the defining polynomial of the elliptic curve in Exercise 2.97(e) is irreducible. Prove the same for the hyperelliptic curve of Exercise 2.97(f). [H]
2.99Show that for an ideal the following two conditions are equivalent:

  1. is generated by a set of homogeneous polynomials.

  2. If , where fi is the sum of non-zero terms of degree i in f, then for all i = 0, . . . , d. (The polynomials fi are called the homogeneous components of f.)

An ideal satisfying the above equivalent conditions is called a homogeneous ideal. Construct an example to demonstrate that all ideals of K[X1, . . . , Xn] need not be homogeneous.

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