on rational points of the third degree thue equationegoins/notes/on_large_rational_solutions_of...on...

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…

1Monday, November 30, 2009

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


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


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.


Finding Solutions to the Cubic Thue Equation

• Integer solutions: (1,0), (2,1)

Infinitely Many!


Finding Rational Solutions

First we must see if there are infinitely many rational solutions.

Rational

N = 1,2 None/Infinite (Pell)

N = 3 Unknown

d = 2 Finite

d = 7 Infinite

N > 3 Finite (Faltings)

Integer

N=1,2 Infinite (Pell)

N > 2 Finite (Thue)


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.


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.


Norm and Conjugate

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.

Lemma:


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


There exists a unique , where such that for each element there exists such that .

Uniqueness of Fundamental Solution

Fix d>0 and G as before.

is called the fundamental solution of .

Proposition:


Let . Consider the the following identities:

and

Assume . Let be the smallest element such that

. Choose .

Uniqueness of Fundamental Solution

Sketch of Proof:


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:


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:

Continued Fractions


Example

If h is even, then and so

.

Let d=6, then we have ,

so h=2 is even. So and

Example:


ExampleIf h is odd, then and so

.

• d=61, • h=11 is odd

Example:

•

•


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.



Proof:

• Let and .

•

• Note that , but so as ,

and . Hence .

• As ,



Let .

Hence .


Axel Thue’s Equation

Thue’s Equation:

If N=3, we have

such that the discriminant


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.


Elliptic Curves

PQ P*Q

such that

P = (x,y) [-1]P = (x,-y)


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


Transformations for Cubic Thue Equation with a Rational Flex Point (u0,v0)

This gives a birational transformation to where .

where w0 satisfies


Example

where a = m = -1, b = c = 0•

•

• C transforms to

Transformation between (u,v) and (x,y) reduces to

} {25Monday, November 30, 2009

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.


Properties of Sequences of Large Rational Points

Plugging x into the 3-division polynomial proves that

is a point of order 3.

Proof:


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.


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)


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:


Table:

[q] [q](x,y) (u,v) u/v3 (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, -55.0912)

1.912930638

159 (44.0055, -253.0765) (469.1832, 245.2693) 1.912931189


Occurrence of Cubics with Rational Points of Inflection

0.16%


Rational Substitution

Given a rational point (u,v) on

substitute

then (X,Y) is on the elliptic curve where .


The Isogeny between E and E’

with dual map} {


Algorithm for Thue Equation with No Rational Flex Points

1. Transform C to E’2. Calculate for E’3. Find a sequence of convergents of4. Compute [qn](x1, y1)5. Transform E’ to C


Example

is isogenous to

Mordell-Weil group is finite with generator:


Example

is isogenous to

Mordell-Weil group is finite with generator (8,24)


Example

is isogenous to

with Rank 1

[12] (28,80) gives one “Large” Rational Point.

Mordell-Weil group is generated by (28,80)


Ranks and Torsion Subgroups

d


Ranks and Torsion Subgroups

• mwrank, PARI/GP, apecs, Maple

• About 63.0% of d values have positive rank.


Thue Equations with Flex Points


Thue Equations with Flex Points

• 3.76% has flex points,

• 0.16% has flex points,

• Algorithm works only for rank > 0.

is an integer.


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.


Acknowledgements

• Edray Goins, Research Seminar Director• Lakeshia Legette, Number Theory

Graduate Assistant• SUMSRI, especially Sara Blight • National Security Agency• National Science Foundation


Questions

Remember, there are no stupid questions…

Just stupid people!


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