On Rational Points of the Third Degree Thue Equationegoins/notes/On_Large_Rational_Solutions_of...On...
Transcript of 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…
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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
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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
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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.
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Finding Solutions to the Cubic Thue Equation
• Integer solutions: (1,0), (2,1)
Infinitely Many!
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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)
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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.
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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.
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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:
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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
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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:
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Let . Consider the the following identities:
and
Assume . Let be the smallest element such that
. Choose .
Uniqueness of Fundamental Solution
Sketch of Proof:
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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:
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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
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Example
If h is even, then and so
.
Let d=6, then we have ,
so h=2 is even. So and
Example:
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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.
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Sequence of Large Rational Solutions
Proof:
• Let and .
•
• Note that , but so as ,
and . Hence .
• As ,
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Axel Thue’s Equation
Thue’s Equation:
If N=3, we have
such that the discriminant
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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.
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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
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Transformations for Cubic Thue Equation with a Rational Flex Point (u0,v0)
This gives a birational transformation to where .
where w0 satisfies
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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.
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Properties of Sequences of Large Rational Points
Plugging x into the 3-division polynomial proves that
is a point of order 3.
Proof:
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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.
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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)
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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:
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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
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Rational Substitution
Given a rational point (u,v) on
substitute
then (X,Y) is on the elliptic curve where .
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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
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Example
is isogenous to
Mordell-Weil group is finite with generator (8,24)
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Example
is isogenous to
with Rank 1
[12] (28,80) gives one “Large” Rational Point.
Mordell-Weil group is generated by (28,80)
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Ranks and Torsion Subgroups
• mwrank, PARI/GP, apecs, Maple
• About 63.0% of d values have positive rank.
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Thue Equations with Flex Points
• 3.76% has flex points,
• 0.16% has flex points,
• Algorithm works only for rank > 0.
is an integer.
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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.
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Acknowledgements
• Edray Goins, Research Seminar Director• Lakeshia Legette, Number Theory
Graduate Assistant• SUMSRI, especially Sara Blight • National Security Agency• National Science Foundation
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