Introduction. Axel Thue was a Mathematician. John Pell was a Mathematician. Most of the people in the audience are Mathematicians.

Transcription

1 Introduction Axel Thue was a Mathematician. John Pell was a Mathematician. Most of the people in the audience are Mathematicians. Giving the Number Theory Group the title 1

2 On Rational Points of the Third Degree Thue Equation What Thue did to Pell! by: Jarrod Cunningham Nancy Ho Karen Lostritto Jon Middleton Nikia Thomas 2

3 John Pell Born in England in Studied Number Theory and Algebra. Pell s Equation: First studied by Brahmagupta, an Indian Mathematician, many years before Pell; but Euler attributed the equation to Pell because Pell wrote a book on it. Pell s Equation has infinitely many integer (when d > 0) and rational solutions. It is also known that as 3

4 Axel Thue Born in Norway in Applied Mathematician. He is famous for proving that there are finitely many integer solutions to the equation when N > 2. 4

5 Finding Solutions to the Cubic Thue Equation Integer solutions: (1,0), (2,1) Infinitely Many! 5

6 Finding Rational Solutions First we must see if there are infinitely many rational solutions. Integer N=1,2 Infinite (Pell) N > 2 Finite (Thue) Rational N = 1,2 None/Infinite (Pell) N = 3 Unknown d = 2 Finite d = 7 Infinite N > 3 Finite (Faltings) 6

7 Finding Large Rational Solutions There are already programs to determine if a cubic Thue equation has infinitely many rational solutions. Assume we have a cubic with infinitely many rational solutions. How do we find large rational solutions to this equation? In this talk, we will discuss an algorithm to generate an infinite sequence of large rational solutions using elliptic curves. We will also exhibit, as an application, that large rational solutions give an approximation of the cube root of d. 7

8 Pell s Equation Pell s Equation: Fix a non-square. Then Consider the ring of algebraic integers Denote as the conjugate of a, and denote as the norm of a. 8

9 Norm and Conjugate Lemma: If d is not a square, then both the conjugate and the norm of a are well defined. Example: Let d=1. Then Hence the conjugate of a is not well-defined. 9

10 Pell s Equation Consider the set. It follows that if, then. Note that G is an abelian group under multiplication. Given two elements, we have 10

11 Uniqueness of Fundamental Proposition: Fix d>0 and G as before. Solution There exists a unique, where such that for each element there exists such that. is called the fundamental solution of. 11

12 Uniqueness of Fundamental Sketch of Proof: Solution Let. Consider the the following identities: and Assume. Let be the smallest element such that. Choose. 12

13 Continued Fractions The fundamental solution can be found using continued fractions. Given a real number x, define the sequence in terms of the floor function, where x 0 =x. We define the continued fraction of x by : Denote and use the notation: 13

14 Continued Fractions Continued fractions of the square root of a squarefree integer is of the form: and is periodic. Let h denote the number of terms that repeat indefinitely. Consider the h th convergent: 14

15 Example If h is even, then. and so Example: Let d=6, then we have, so h=2 is even. So and 15

16 Example If h is odd, then Example: d=61, h=11 is odd. and so 16

17 Theorem: Sequence of Large Rational Solutions Say is a fundamental solution. Denote for n=0,1,2,. As, Moreover, the ratio, as. Note that the theorem is false if d is negative. 17

18 Proof: Sequence of Large Rational Solutions Let and. Note that, but so as, and. Hence. As, 18

19 Sequence of Large Rational Solutions Let. Hence. 19

20 Axel Thue s Equation Thue s Equation: If N=3, we have such that the discriminant 20

21 Thue s Equation with Rational Points of Inflection We will later show that if C has a rational point of inflection, then it will be birationally equivalent to an elliptic curve. 21

22 Elliptic Curves such that P Q P*Q P = (x,y) [-1]P = (x,-y) 22

23 Elliptic Curves E( ) = the collection of rational points forms an abelian group. E( ) tors = collection of points of finite order. Rank = number of generators for E( ) / E( ) tors 23

24 Transformations for Cubic Thue Equation with a Rational Flex Point (u 0,v 0 ) where w 0 satisfies This gives a birational transformation to where. 24

25 Example C transforms to where a = m = -1, b = c = 0 Transformation between (u,v) and (x,y) reduces to } { 25

26 Properties of Sequences of Large Rational Points Theorem: Assume C is a cubic Thue Equation with a rational flex point. A sequence {(u n, v n )} on C such that u n, v n as corresponds to a sequence {(x n,y n )} on E such that as. This limit is a point of order 3. 26

27 Proof: Properties of Sequences of Large Rational Points Plugging x into the 3-division polynomial proves that is a point of order 3. 27

28 Large Rational Solutions Fix an elliptic curve E of the form where D = -16m 2 Disc There exists a group isomorphism: where Define, where (x 1,y 1 ) on E. 28

29 Algorithm for Thue Equations with a Rational Flex Point 1. Find the generator (x 1,y 1 ) of E. 2. Find continued fraction and convergents of (x n, y n ) = [q n ](x 1, y 1 ) has approximate order Find the sequence [q n ](x 1,y 1 ) where 3 p n. 4. Transform (x n,y n ) on E to (u n,v n ) on C. Proof: Define P = [q](x 1,y 1 ) 29

30 Example: Finding Large Rational Points a = m = -1, b = c = 0, d = 7 C is birationally equivalent to where Find convergents of such that p n is not divisible by 3: 30

31 Table: [q] [q](x,y) (u,v) u/v 3 (57, -405) (4.2941, ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

32 Occurrence of Cubics with Rational Points of Inflection 0.16% 32

33 Rational Substitution Given a rational point (u,v) on substitute then (X,Y) is on the elliptic curve where. 33

34 The Isogeny between E and E } { with dual map 34

35 Algorithm for Thue Equation with No Rational Flex Points 1. Transform C to E 2. Calculate for E 3. Find a sequence of convergents of 4. Compute [q n ](x 1, y 1 ) 5. Transform E to C 35

36 Example is isogenous to Mordell-Weil group is finite with generator: 36

37 Example is isogenous to Mordell-Weil group is finite with generator (8,24) 37

38 Example is isogenous to with Rank 1 Mordell-Weil group is generated by (28,80) [12] (28,80) gives one Large Rational Point. 38

39 Ranks and Torsion Subgroups d 39

40 Ranks and Torsion Subgroups mwrank, PARI/GP, apecs, Maple About 63.0% of d values have positive rank. 40

41 Thue Equations with Flex Points 41

42 Thue Equations with Flex Points 3.76% has flex points, 0.16% has flex points, Algorithm works only for rank > 0. is an integer. 42

43 Future Research Using the Cubic Thue Equation, is there a pattern to predict which d s give you finitely or infinitely many solutions? If C doesn t have a rational point of inflection, how well does our algorithm work for finding large rational solutions? Because the map from C to E is not surjective, more work is necessary to determine how much information rational points on E will give us about rational points on C. 43

44 Acknowledgements Edray Goins, Research Seminar Director Lakeshia Legette, Number Theory Graduate Assistant SUMSRI, especially Sara Blight National Security Agency National Science Foundation 44

45 Questions Remember, there are no stupid questions Just stupid people! 45