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… Monday, November 30, 2009 1 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 Monday, November 30, 2009 2 John Pell • Born in England in 1611. • 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 Monday, November 30, 2009 3 Axel Thue • Born in Norway in 1863. • Applied Mathematician. • He is famous for proving that there are finitely many integer solutions to the equation when N > 2. Monday, November 30, 2009 4 Finding Solutions to the Cubic Thue Equation • Integer solutions: (1,0), (2,1) Infinitely Many! Monday, November 30, 2009 5 Finding Rational Solutions First we must see if there are infinitely many rational solutions. Integer Rational N=1,2 Infinite (Pell) N = 1,2 None/Infinite (Pell) N > 2 Finite (Thue) N = 3 Unknown d = 2 Finite d = 7 Infinite N > 3 Finite (Faltings) Monday, November 30, 2009 6 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. Monday, November 30, 2009 7 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. Monday, November 30, 2009 8 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. Monday, November 30, 2009 9 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 Monday, November 30, 2009 10 Uniqueness of Fundamental Solution Proposition: Fix d>0 and G as before. There exists a unique , where such that for each element there exists such that . is called the fundamental solution of . Monday, November 30, 2009 11 Uniqueness of Fundamental Solution Sketch of Proof: Let . Consider the the following identities: and Assume . Let be the smallest element such that . Choose . Monday, November 30, 2009 12 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 x0=x. •We define the continued fraction of x by : •Denote and use the notation: Monday, November 30, 2009 13 Continued Fractions Continued fractions of the square root of a square- free integer is of the form: and is periodic. Let h denote the number of terms that repeat indefinitely. Consider the hth convergent: Monday, November 30, 2009 14 Example If h is even, then and so . Example: Let d=6, then we have , so h=2 is even. So and Monday, November 30, 2009 15 Example If h is odd, then and so . Example: • d=61, • h=11 is odd • • Monday, November 30, 2009 16 Sequence of Large Rational Solutions Theorem: 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. Monday, November 30, 2009 17 Sequence of Large Rational Solutions Proof: • Let and . • • Note that , but so as , and . Hence . • As , Monday, November 30, 2009 18 Sequence of Large Rational Solutions Let . Hence . Monday, November 30, 2009 19 Axel Thue’s Equation Thue’s Equation: If N=3, we have such that the discriminant Monday, November 30, 2009 20 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. Monday, November 30, 2009 21 Elliptic Curves such that Q P*Q P P = (x,y) [-1]P = (x,-y) Monday, November 30, 2009 22 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 Monday, November 30, 2009 23 Transformations for Cubic Thue Equation with a Rational Flex Point (u0,v0) where w0 satisfies This gives a birational transformation to where . Monday, November 30, 2009 24 Example • where a = m = -1, b = c = 0 • C transforms to • Transformation between (u,v) and (x,y) reduces to } { Monday, November 30, 2009 25 Properties of Sequences of Large Rational Points Theorem: Assume C is a cubic Thue Equation with a rational flex point. A sequence {(un, vn)} on C such that |un|, |vn| as corresponds to a sequence {(xn,yn)} on E such that as . This limit is a point of order 3. Monday, November 30, 2009 26 Properties of Sequences of Large Rational Points Proof: Plugging x into the 3-division polynomial proves that is a point of order 3. Monday, November 30, 2009 27 Large Rational Solutions Fix an elliptic curve E of the form where D = -16m2 Disc There exists a group isomorphism: where Define , where (x1,y1) on E. Monday, November 30, 2009 28 Algorithm for Thue Equations with a Rational Flex Point 1. Find the generator (x1,y1) of E. 2. Find continued fraction and convergents of (xn, yn) = [qn](x1, y1) has approximate order 3. 3. Find the sequence [qn](x1,y1) where 3 | pn . 4. Transform (xn,yn) on E to (un,vn) on C. Proof: Define P = [q](x1,y1) Monday, November 30, 2009 29 Example: Finding Large Rational Points a = m = -1, b = c = 0, d = 7 C is birationally equivalent to where Find convergents of such that pn is not divisible by 3: Monday, November 30, 2009 30 Table: [q] [q](x,y) (u,v) u/v 3 (57, -405) (4.2941, 2.2353) 1.921052631 7 (42.0481, -230.5966) (-22.5476, -11.7873) 1.912875562 121 (43.4989, -247.2625) (-105.3857, 1.912930638 -55.0912) 159 (44.0055, -253.0765) (469.1832, 245.2693) 1.912931189 Monday, November 30, 2009 31 Occurrence of Cubics with Rational Points of Inflection 0.16% Monday, November 30, 2009 32 Rational Substitution Given a rational point (u,v) on substitute then (X,Y) is on the elliptic curve where . Monday, November 30, 2009 33 The Isogeny between E and E’ } with dual map { Monday, November 30, 2009 34 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 [qn](x1, y1) 5. Transform E’ to C Monday, November 30, 2009 35 Example is isogenous to Mordell-Weil group is finite with generator: Monday, November 30, 2009 36 Example is isogenous to Mordell-Weil group is finite with generator (8,24) Monday, November 30, 2009 37 Example is isogenous to with Rank 1 Mordell-Weil group is generated by (28,80) [12] (28,80) gives one “Large” Rational Point. Monday, November 30, 2009 38 Ranks and Torsion Subgroups d Monday, November 30, 2009 39 Ranks and Torsion Subgroups • mwrank, PARI/GP, apecs, Maple • About 63.0% of d values have positive rank. Monday, November 30, 2009 40 Thue Equations with Flex Points Monday, November 30, 2009 41 Thue Equations with Flex Points • 3.76% has flex points, • 0.16% has flex points, • Algorithm works only for rank > 0. is an integer. Monday, November 30, 2009 42 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. Monday, November 30, 2009 43 Acknowledgements • Edray Goins, Research Seminar Director • Lakeshia Legette, Number Theory Graduate Assistant • SUMSRI, especially Sara Blight • National Security Agency • National Science Foundation Monday, November 30, 2009 44 Questions Remember, there are no stupid questions… Just stupid people! Monday, November 30, 2009 45.