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Torsion Points of Elliptic Curves Over Number Fields

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Torsion Points of Elliptic Curves Over Number Fields Torsion Points of Elliptic Curves Over Number Fields Christine Croll A thesis presented to the faculty of the University of Massachusetts in partial fulfillment of the requirements for the degree of Bachelor of Science with Honors. Department of Mathematics Amherst, Massachusetts April 21, 2006 Acknowledgements I would like to thank my advisor, Prof. Farshid Hajir of the University of Mas- sachusetts at Amherst, for his willingness to explain everything twice. I would also like to thank Prof. Tom Weston of the University of Massachusetts at Amherst, for entertaining me during my 9 a.m. Number Theory course; Prof. Peter Norman of the University of Massachusetts, for offering me the summer research that inspired this thesis; and the Umass Mathematics department for my Mathematical education. Lastly, to the English major, Animal Science major, and Biology major that have had to live with me, thank you for putting up with this strange Math major. Abstract A curve C over Q is an equation f(x, y) = 0, where f is a polynomial: f ∈ Z[x, y]. It is interesting to study the set of rational points for curves, denoted as C(Q), which consists of pairs (x, y) ∈ Q2 satisfying f(x, y) = 0. For curves of degree 1 and 2 we know how to write C(Q) as a parametrized set, therefore enabling us to know all the rational points for curves of these degrees. For irreducible, smooth curves of degree 3 equipped with at least one rational point, which are called elliptic curves, C(Q) is a group. As with other groups, each element of C(Q) has an order. The subgroup C(Q)tors is the set of all rational points of finite order in C(Q). This paper will briefly describe the basic theory of elliptic curves and then focus on what is known about C(Q)tors, both over the rationals and over the extension field Q(ζ11), where ζ11 is an irrational root of x11 − 1 = 0. Contents 1 Introduction 5 2 Background 8 2.1 Curves . 8 2.2 Rational Points on Conics . 8 2.3 Method for Rational Solutions for Quadratics . 9 2.4 Projective Space . 11 3 Elliptic Curves 13 3.1 EC Definitions . 13 3.2 Weierstrass Form . 13 3.3 EC Definitions Revisited . 13 3.4 Projective Space Revisited . 14 4 Group Law 15 4.1 Defining ⊕ ............................. 15 4.2 (P ∗ Q) ⊕ (P ⊕ Q) ........................ 16 4.3 P ⊕ P............................... 17 4.4 Elliptic Curves Under ⊕ ..................... 19 5 Torsion Points 20 5.1 Definition of torsion points . 20 5.2 Points of Order 2 . 20 5.3 Points of Order 3 . 21 5.4 Nagell-Lutz Theorem . 22 5.5 Mazur’s Theorem . 23 5.6 Ψn(x) ............................... 24 6 Appendix A 26 7 Appendix B 27 8 Appendix C 28 5 1 Introduction The subject of rational points on elliptic curves could be seen as a component of the theory of Diophantine equations. The study of elliptic curves has come a long way since its beginning, with elliptic curves currently being used in the area of cryptography. My research did not delve into the role of elliptic curves in cryptography; instead I studied basic theoretical tools for understanding rational points on an elliptic curve, which, simply put, are solutions with co- ordinates which can be expressed as ratios of whole numbers. Interestingly enough, for a certain geometrically-defined binary operation ⊕, to be described in more detail in chapter 4, we can turn the set of rational points, notated as C(Q), into a commutative group. Furthermore, we can classify elements into torsion points and non-torsion points, which are points of finite order and infinite order, respectively. In considering the set of all torsion points, defined as C(Q)tors, we find that this set forms a subgroup. It is actually known that for an elliptic curve C over Q that contains a point of finite order m, either 1 ≤ m ≤ 10 or m = 12. More precisely, the set of all points of finite order in C(Q) forms a subgroup which has one of the following two forms: (i) A cyclic group of order N with 1 ≤ N ≤ 10 or N = 12. (ii) T he product of a cyclic group of order two and a cyclic group of order 2N with 1 ≤ N ≤ 4. This characterization of C(Q)tors is known as Mazur’s Theorem. Another valuable theorem is the Nagell-Lutz Theorem. Theorem 1 Let y2 = f(x) = x3 + ax2 + bx + c be a non-singular cubic curve with integer coefficients a,b,c; and let D be the dis- criminant of the cubic polynomial f(x), D = −4a3c + a2b2 + 18abc − 4b3 − 27c2. Let P = (x,y) be a rational point of finite order. Then x and y are inte- gers; and either y = 0, in which case P has order two, or else y divides D. 6 Once our cubic is put in the proper form, the Nagell-Lutz Theorem gives us a procedure, although one that can be long and tedious, for finding the points of finite order for an elliptic curve. The procedure is simple; take D and find all the integers that divide it. These are all of the possible y values. From here we plug in each y value into our curve and factor the resulting cubic to obtain any integer x values. The bigger D is, the longer this process takes. Luckily there is a stronger version of Nagell-Lutz which tells us that y2 divides D, not just y. This means we only need to look at which perfect squares divide D instead of which integers. This significantly cuts down on the number of y values we have to process. After listing all possible (x, y) satisfying the hypotheses of the theorem, we then still need to check which of these are actually torsion points. In addition to having a method for finding all the rational torsion points for a given curve, there exists a way to find all the torsion points of a certain order for that curve. For each integer n, there exists a polynomial with ra- tional coefficients called Ψn(x) where the roots of this polynomial are the x values of the torsion points of order n. For the rational torsion points of order n you simply restrict your view to the roots of Ψn(x) that are rational. For example, to find all the torsion points of order 3 on the curve y2 = 3 4 2 2 x + bx + c you would use Ψ3(x) = 3x + 6bx + 12cx − b . Finding the roots of Ψ3 will give us the x-coordinates of our torsion points. To find the correspond- ing y values you simply plug these x values into our elliptic curve and solve for y. Note that because our elliptic curve contains a y2 term, we will get two y values for every x value. In other words, every x value yields two torsion points. Remember, Ψn will give you all the possible torsion points of order n, both rational and irrational. If Ψn has no rational roots, then there will be no ra- tional torsion points of order n. Although the study of elliptic curves over Q is very rich, the topic broadens even further when you consider elliptic curves over algebraic number fields. An algebraic number field is a finite field extension of the rational numbers. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. In other words, take Q and adjoin to it a root of a polynomial that is not found in Q. This gives us a field that contains a copy of Q and all linear combinations of powers of our adjoined root. 2 3 2 I focused on the curve X0(11) which is y = x −4x −160x−1264 over the 7 11 algebraic number field Q(ζ11), where ζ11 is an irrational root of x − 1 = 0, i.e. a root of x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 1 = 0. In considering the size of the torsion subgroup of X0(11) over Q(ζ11), I discovered the subgroup has cardinality 5, the same cardinality as the tor- sion subgroup over Q. In fact, the same 5 points make up the two torsion subgroups. Therefore, for X0(11), the torsion subgroup was not effected by looking for points over Q(ζ11) instead of Q. 8 2 Background 2.1 Curves Before we can talk about elliptic curves, we must first review a few basic definitions, starting with the definition of a curve. A curve C, is an equa- tion f(x, y) = 0, where f is a polynomial: f ∈ Z[x, y]. Without further comment, we will always assume that our curves are irreducible (meaning f is an irreducible polynomial) and smooth, meaning the system of equations f(x, y) = 0, ∂xf = 0, ∂yf = 0 has no solutions. A rational solution of a curve is a pair (x, y) such that x, y ∈ Q and f(x, y) = 0 . Similarly, an integer solution of a curve is a pair (x, y) such that x, y ∈ Z and f(x, y) = 0. The definitions of rational and integer solutions give rise to two natural subsets of our curve, called C(Q) and C(Z). C(Q) := {(x, y) ∈ Q × Q | f(x, y) = 0} and C(Z) := {(x, y) ∈ Z × Z | f(x, y) = 0} . The majority of this paper will focus on C(Q).
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