Rational Points on Curves (2006)

Teacher: F.Beukers

Literature

M.Hindry, J.Silverman, Diophantine Geometry, Springer Verlag. We will follow chapters from this book. Unfortunately this book is out of print, but fortunately it will be in the Springer Yellow Sale starting March 1. I placed an order on behalf of our class.
R. Hartshorne, Algebraic Geometry, Springer Verlag. We will use some of the exercises in Chapter 1.
For a background in algebraic number theory we recommend the notes of the 2004 Master Course Algebraic Number Theory, first two chapters. For a bit more reference on algebraic geometry we refer to the notes of the 2005 National Master Course Elliptic Curves, Chapters 1,,3,5,6,8.

Weekly schedule:

Feb 7
Introduction to diophantine equations.
Start of Section 1.1 of Hindry/Silverman.
Homework for next time: Problems 2), 3) assigned in class and Exercises 1.1, 1.3 from Hartshorne. Note that use is made of the affine coordinate ring, which we did not have time to define, but it is simply the polynomial ring k[X_1,...,X_n] modulo the defining ideal (page 4 of Hartshorne or page 11 of Hindry/Silverman).
solutionsrationalpoints1.pdf.
Feb 14
Introduction to terminology from algebraic geometry.
Afiine and projective varities, singularities, morphisms and rational maps.
Homework for next time:
1) For which values of l does the projective cubic surface
x³ + y³ + z³ + u³ = l (x + y + z + u)³
have singularities? (In one instance there are four singularities, the largest number of isolated singularities for a cubic surface)
2) Write x=(x₀, x₁, x₂, x₃) and let A be a symmetric four by four matrix. Consider the intersection of the quadratic hypersurfaces in 3-dimensional projective space
x.x = 0, x.A.x = 0.
We assume that A is not a scalar multiple of the identity matrix. Give conditions on A so that the curve of intersection is singular. (Hint: start with diagonal A).
solutionsrationalpoints2.pdf
Feb 21
Regular functions and regular maps. Examples and special attention to rational functions on curves.
Home work for next time:
1) We provide P⁵ with the six homogeneous coordinates X_ij i=0,1; j=0,1,2. Denote the homogeneous coordinates on P¹ by x₀, x₁ and the homogeneous coordinates on P² by y_j, j=0,1,2. Show that the image of the Segre embedding P¹ x P² to P⁵ given by X_ij = xⁱy^j, i=0,1; j=0,1,2 is a projective variety given by the equations X_ijX_kl = X_kjX_il for all i,k=0,1 and j,l=0,1,2. Hint: consider the coordinates X_ij as entries of 2 by 3 matrix.
2) a) Show that the image Q of the Segre map P¹ x P¹ to P³ is given by the equation xv - yu = 0, where x,y,u,v are the homogeneous coordinates in P³.
2 b) Show that Q is birationally equivalent to P².
2 c) Show that Q is not isomorphic to P². Hint: it is given that any two projective curves in P² have a point of intersection.
3) Recall that a line through two points a=(a₀:a₀:...:a_n) and b=(b₀:b₀:...:b_n) is given by the homogeneous parameter representation la+mb with homogeneous parameters l,m. a) Consider the rational projection map from P³ to P² given as follows. The coordinates of P³ are denoted (x:y:z:w). Take any point (x:y:z:w) and take the line through this point and P=(0:0:0:1). Intersect this line with hyperplane w = 0. Show that this gives the map (x:y:z:w) to (x:y:z). We call this the projection from P onto the hyperplane w = 0. On which points is this map a morphism? b) Now consider the algebraic curve given by the equations x²-xz-yw=0 and yz-xw-zw=0. Show that the projection from a) gives a morphism of this curve minus P onto the cubic plane curve y²z-x³+xz²=0. (This an expanded version of Exercise 5.11 from Hartshorne)
4) Do exercises 5.1 and 5.2 from Hartshorne.
solutionsrationalpoints3.pdf
Feb 28
Algebraic curves, normalisation, local parameters, zeros and poles of rational functions, differential forms.
Home work for next Monday:
1) Let C be a plane affine curve in A² (with coordinates x,y). Suppose that (0,0) is a smooth point of C and suppose that the tangent line of C at (0,0) is not the line y=0.

a) Show that the equation of C can be written in the form x + xQ(x,y) + q(y) = 0 where Q is in k[x,y] and Q(0,0)=0, and q is in k[y] and q(0)=0. (Note: you can multiply your equation by a non-zero constant to normalise the coefficient of x if necessary).
b) Let O_(0,0) be the local ring of all f in k(C), regular in (0,0) and let M_(0,0) be the ideal in O_(0,0) consisting of all f that vanish in (0,0). Show that M_(0,0) is generated by the function y (i.e. y is a local parameter at (0,0)).
c) Show that to any rational function f in k(C) there exists an integer n and a rational function g in O_(0,0), invertible in O_(0,0), such that f = yⁿg. (Hint: first do this problem for the function x).

2) Consider the plane projective curve C given by x³y + y³z + z³x = 0. (Attention: the last term in this polynomial was misstated, I corrected this to z³x). Let f be the rational function on C given by x/z.
a) Determine the degree of f. (For any finite dominant map f: X->Y between two varities X,Y the degree is defined as the degree of the function field extension k(X)/f*k(Y), where f*:k(Y)->k(X) is the natural embedding of function field initiated by f. Finiteness and dominance of a rational map f:C->D between curves C,D is garanteed as soon as the map f is non-constant).
b) Determine the zeros and poles of f, together with their multiplicities.
c) Determine all points P on C where f has index of ramification e_P>1. (Hint: determine b such that x/z=b has multiple solutions on C by elimination of x)
solutionsrationalpoints4.pdf
March 7
Differential forms, divisors, canonical divisor, Riemann-Roch theorem for curves.
First assignment (due March 21).
In all of the questions we consider our curves over an algebraically closed field k of characteristic zero. We use Hindry/Silverman section A4 as main reference.
- (1) Consider the cubic curve E: y² = x³+ax+b, where x³+ax+b has distinct zeros. The closure of E in the projective plane contains one extra point, which we denote by O, the point at infinity.
  Without using the Riemann-Roch theorem prove that for every integer n>0 the dimension of L(nO) equals n and display a basis of this space. (Hint: functions in L(nO) are regular on the affine part of E; what are the orders of x,y at O?)
- (2) Let C be a smooth projective algebraic curve. A birational isomorphism from C to itself is called an automorphism. The automorphism group of C is denoted by Aut(C).
  - Suppose the genus of C is 0. Show that Aut(C) is isomorphic to the group GL(2,k) modulo scalars. (Hint: Notice that C is isomorphic to P¹ and that rational functions on P¹ can be considered as rational maps from P¹ to itself)
  - Suppose the genus of C is 1 and write C in standard Weierstrass form y² = x³+ax+b. Show that if ab not equal 0, then Aut(C) is generated by the translations P |-> P+Q with a fixed Q on C and the involution (x,y) |-> (x,-y). (Hint: If s is in Aut(C), then there exists a translation T so that Ts fixes the point at infinity).
    Suppose ab=0, write down a set of generators for Aut(C).
    In these problems you may only use basic definitions and the result of problem (1). So no standard theorems on elliptic curves, you are proving one of them.
- (3) Do Problem A.4.2 from Hindry/Silverman. There is an error in part (d) of the problem. The embedding should be phi: C -> P^[d/2]+1 given by (x,y) |-> (1,x,...,x^[d/2],y), where d=deg(F). Please use this one in the problem. Sorry about the confusion.
March 14
Linear systems corresponding with divisors, very ample divisors, canonical embedding. Beginning of classification of curves over k (k algebraic number field) and their k-rational points.
March 21
Classification of algebraic curves over algebraic number fields. Rational curves, elliptic curves, curves of genus 2 and 3. Overview of main theorems.
March 28
Second Assignment (due April 11)
- (1) Let C,D be plane projective curves given by the homogeneous equations F(x,y,z)=0 and G(x,y,z)=0 respectively. Let n,m be the degrees of F,G respectively. Suppose C and D intersect in a finite number of points. Suppose in addition that each intersection point is a smooth point of both C and D and that the intersection is transversal (i.e. the tangents of C,D at the point of intersection are distinct). Then the Theorem of Bezout asserts that there are precisely mn points of intersection. Assume in addition that the line z=0 intersects C in n distinct points. Prove Bezout's theorem by considering the zeros and poles of the rational function G(x,y,z)/z^m on C.
- (2) Consider the plane projective curve E given by the affine equation y² =f(x) where f(x) is a quartic polynomial with distinct zeros. From problem A.4.2 of Hindry/Silverman we know that E has genus 1. Let P=(x₀,y₀) be a point on E and suppose that y₀ is non-zero. Then, according to the Riemann-Roch theorem, the space L(3P) has dimension 3. We construct this space as follows.
  - Show that there exist a,b,c such that q(x,y)=y+ax²+bx+c has a zero of order 3 at P (Hint:use x-x₀ as local parameter).
  - Show that there exists a fourth zero which we denote by Q.
  - Show that L(3P) coincides with the space of functions p(x,y)/q(x,y) where p(x,y) is a polynomial of the form a'y+b'x²+c'x+d' with the restriction that it vanishes in Q.
- (3) Suppose we want to determine the Q-rational points on the curve E given by y² =2x⁴+x³+2x²+1. We note the rational point P=(0,1). So E is an elliptic curve. Use the construction in problem 2) to give a basis of the space L(3P). Find a birational map E->E' where E' is an elliptic curve in weierstrass normal form Y²=g(X), with g(X) of degree 3, such that P is mapped to the point at infinity. Determine g(X).
- (4) Do problem A.4.9 of Hindry/Silverman
April 4
Proof of the Mordell-Weil theorem for elliptic curves over Q.
April 11
Crash course in algebraic number theory, fields with norms, product formula, height on projective spaces (Hindry/Silverman B.1, B.2) Here you find the answers to the first assignment.
April 18
Schmidt's subspace theorem and the Corvaja-Zannier proof of Siegel's theorem on integral points on curves. Here are the answers to Assignment II
April 25
End of proof of Siegel's theorem, theorem of Chevalley-Weil, Runge's theorem and application of Schmidt's subspace theorem to S-unit equations. Here you find Assignment III, due May 9 To remind you, when k is an algebraic number field and S a finite set of norms on k, including the Archimedean ones, an S-integer in k is an element a in k such that ||a||_v is less than or equal to 1 for all norms v not in S. The element a is called an S-unit if ||a||_v=1 for norms v not in S.
May 2
No class
May 9 (Minnaert 012)
May 16