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18.782 Arithmetic Geometry Lecture Note 1

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18.782 Arithmetic Geometry Lecture Note 1 Introduction to Arithmetic Geometry 18.782 Andrew V. Sutherland September 5, 2013 1 What is arithmetic geometry? Arithmetic geometry applies the techniques of algebraic geometry to problems in number theory (a.k.a. arithmetic). Algebraic geometry studies systems of polynomial equations (varieties): f1(x1; : : : ; xn) = 0 . fm(x1; : : : ; xn) = 0; typically over algebraically closed fields of characteristic zero (like C). In arithmetic geometry we usually work over non-algebraically closed fields (like Q), and often in fields of non-zero characteristic (like Fp), and we may even restrict ourselves to rings that are not a field (like Z). 2 Diophantine equations Example (Pythagorean triples { easy) The equation x2 + y2 = 1 has infinitely many rational solutions. Each corresponds to an integer solution to x2 + y2 = z2. Example (Fermat's last theorem { hard) xn + yn = zn has no rational solutions with xyz 6= 0 for integer n > 2. Example (Congruent number problem { unsolved) A congruent number n is the integer area of a right triangle with rational sides. For example, 5 is the area of a (3=2; 20=3; 41=6) triangle. This occurs iff y2 = x3 − n2x has infinitely many rational solutions. Determining when this happens is an open problem (solved if BSD holds). 3 Hilbert's 10th problem Is there a process according to which it can be determined in a finite number of operations whether a given Diophantine equation has any integer solutions? The answer is no; this problem is formally undecidable (proved in 1970 by Matiyasevich, building on the work of Davis, Putnam, and Robinson). It is unknown whether the problem of determining the existence of rational solutions is undecidable or not (it is conjectured to be so). If we restrict to equations with just 2 variables (plane curves), then this problem is believed to be decidable, but there is no known algorithm for doing so (unless we also restrict the degree to 2). 4 Algebraic curves A curve is an algebraic variety of dimension 1 (defined over a field k). In n-dimensional affine space kn, this means we have a system of n − 1 polynomials in n variables (n + 1 variables in projective space). For the sake of simplicity we will usually work with plane curves (n = 2) defined by a single polynomial equation. But we should remember that not all curves are plane curves (the intersection of two quadric surfaces in 3-space, for example), and we may sometimes need to embed our plane curves in a higher dimensional space in order to remove singularities. While we will often define our curves using affine equations f(x; y) = 0, it is more convenient to work with projective curves defined by homogeneous equations g(x; y; z) = 0 in which every term has the same degree. Such a g can be obtained by homogenizing f (multiplying each term by an appropriate power of z); we then have f(x; y) = g(x; y; 1). 5 We may call points of the form (x : y : 1) affine points. They form an affine plane A2(k) embedded in P2(k). We may call points of the form (x : y : 0) points at infinity. These consist of the points (x : 1 : 0) and the point (1 : 0 : 0), which form the line at infinity, a projective line P1(k) embedded in P2(k). 2 2 S 1 (of course this is just one way to partition P as A P ). The projective plane Definition The projective plane P2(k) is the set of all nonzero triples (x; y; z) 2 k3, modulo the equivalence relation (x; y; z) ∼ (λx, λy; λz) for all λ 2 k∗: The projective point (x : y : z) is the equivalence class of (x; y; z). 6 The projective plane Definition The projective plane P2(k) is the set of all nonzero triples (x; y; z) 2 k3, modulo the equivalence relation (x; y; z) ∼ (λx, λy; λz) for all λ 2 k∗: The projective point (x : y : z) is the equivalence class of (x; y; z). We may call points of the form (x : y : 1) affine points. They form an affine plane A2(k) embedded in P2(k). We may call points of the form (x : y : 0) points at infinity. These consist of the points (x : 1 : 0) and the point (1 : 0 : 0), which form the line at infinity, a projective line P1(k) embedded in P2(k). 2 2 S 1 (of course this is just one way to partition P as A P ). 7 Plane projective curves Definition A plane projective curve C=k is defined by a homogeneous polynomial f(x; y; z) with coefficients in k.1 For any field K containing k, the K-rational points of Cf form the set 2 C(K) = f(x : y : z) 2 P (K) j f(x; y; z) = 0g: @f @f @f A point P 2 C(K) is singular if @x , @y , @z all vanish at P . C is smooth (or nonsingular) if there are no singular points in C(k¯). C is (geometrically) irreducible if f(x; y; z) does not factor over k. We will always work with curves that are irreducible. We will usually work with curves that are smooth. 1Fine print: up to scalar equivalence and with no repeated factors in k¯[x; y; z]: 8 Examples of plane projective curves over Q affine equation f(x; y; z) points at 1 y = mx + b y − mx − bz (1 : m : 0) x2 − y2 = 1 x2 − y2 − z2 (1 : 1 : 0), (1; −1; 0) y2 = x3 + Ax + B y2z − x3 − Axz2 − Bz3 (0 : 1 : 0) x4 + y4 = 1 x4 + y4 − z4 none x2 + y2 = 1 − x2y2 x2z2 + y2z2 − z4 + x2y2 (1 : 0 : 0); (0 : 1 : 0) These curves are all irreducible. The first four are smooth (provided that 4A3 + 27B2 6= 0). The last curve is singular (both points at infinity are singular). 9 Genus Over C, an irreducible smooth projective curve is a connected compact C-manifold of dimension 1 (a compact Riemann surface). Every such manifold is topologically equivalent to a sphere with handles. The number of handles is the (topological) genus. genus 0 genus 1 genus 2 genus 3 Later in the course we will give an algebraic definition of genus that works over any field (and agrees with the topological genus over C). The genus is the single most important invariant of a curve. 10 Newton polygon Definition P i j The Newton polygon of a polynomial f(x; y) = aijx y is the 2 convex hull of the set f(i; j): aij 6= 0g in R . An easy way to compute the genus of a (sufficiently general) irreducible curve defined by an affine equation f(x; y) = 0 is to count the integer lattice points in the interior of its Newton polygon:2 y2 = x3 + Ax + B. 11 2See http://arxiv.org/abs/1304.4997 for more on Newton polygons. Birational equivalence A rational map between two varieties is a map defined by rational functions (which need not be defined at every point). For example, the map that sends the point (0; t) on the vertical line x = 0 to the point 1 − t2 2t ; 1 + t2 1 + t2 on the unit circle x2 + y2 = 1 is a rational map of plane curves. The inverse map sends (x; y) on the unit circle to (0; y=(x + 1)). Varieties related by rational maps φ and ' such that both φ ◦ ' and ' ◦ φ are the identity at all points where both are defined are birationally equivalent. The genus is invariant under birational equivalence (as are most things). We will give a finer notion of equivalence (isomorphism) later in the course. 12 In the case of a plane conic ax2 + bxy + cy2 + dx + ey + f = 0 over a field whose characteristic is not 2, after a linear change of variables (complete the square at most 3 times) we obtain an affine equation of the form ax2 + by2 = 1: If this equation has a rational solution (x0; y0), then the substitution x = (1 − x0u)=u and y = (t − y0u)=u yields 2 2 2 2 2 2 a(1 − 2x0u + x0u ) + b(t − 2y0tu + y0u ) = u 2 a + bt − (2ax0 + 2by0t)u = 0; 2 and the parameterizationt; g(t), where g(t) = a+bt 2 k(t). 2ax0+2by0t Curves of genus 0 As may be seen by their Newton polytopes, plane curves C of genus 0 are (in general) either conics (degree 2 curves), or of the form y = g(x). In the latter case we can immediately parameterize the points of C(k): the affine points are all of the form t; g(t), with t 2 k. 13 Curves of genus 0 As may be seen by their Newton polytopes, plane curves C of genus 0 are (in general) either conics (degree 2 curves), or of the form y = g(x). In the latter case we can immediately parameterize the points of C(k): the affine points are all of the form t; g(t), with t 2 k. In the case of a plane conic ax2 + bxy + cy2 + dx + ey + f = 0 over a field whose characteristic is not 2, after a linear change of variables (complete the square at most 3 times) we obtain an affine equation of the form ax2 + by2 = 1: If this equation has a rational solution (x0; y0), then the substitution x = (1 − x0u)=u and y = (t − y0u)=u yields 2 2 2 2 2 2 a(1 − 2x0u + x0u ) + b(t − 2y0tu + y0u ) = u 2 a + bt − (2ax0 + 2by0t)u = 0; 2 and the parameterizationt; g(t), where g(t) = a+bt 2 k(t).
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