elliptic curves: a surveyegoins/notes/4-covering...july 17, 2008 sumsri number theory seminar...

Motivating QuestionsBasic Properties

Mordell-Weil GroupConclusion

Elliptic Curves: A Survey

SUMSRI Number Theory Seminar

Samuel IvyMorehouse College

Brett JeffersonMorgan State University

Michele JoseyNorth Carolina Central University

July 17, 2008

SUMSRI Number Theory Seminar Elliptic Curves: A Survey



Outline

1 Motivating Questions

2 Basic PropertiesElliptic CurvesChord-Tangent MethodGroup Law

3 Mordell-Weil GroupPoincare’s Conjecture / Mordell’s TheoremTorsion SubgroupMordell-Weil Rank

4 Conclusion




Rational Triangles

Can you find

a right triangle

with rational sides

having area A = 6?




Rational Triangles

Consider rational numbers a, b, and c satisfying

a2 + b2 = c2 and1

2a b = 6.

a

b

c

Recall the (a, b, c) = (3, 4, 5) triangle!




Cubic Equations

Are there more rational solutions (a, b, c) to

a2 + b2 = c2 and1

2a b = 6?

Proposition

Let X and Y be rational numbers, and denote the rational numbers

a =X 2 − 36

Y, b =

12 X

Y, and c =

X 2 + 36

Y.

Then

a2 + b2 = c2

1

2a b = 6

if and only if

{Y 2 = X 3 − 36 X .

Example: (X , Y ) = (12, 36) corresponds to (a, b, c) = (3, 4, 5).

Can we find infinitely many rational solutions (a, b, c)?




What types of properties

does this

cubic equation have?




Elliptic CurvesChord-Tangent MethodGroup Law

What is an Elliptic Curve?

Definition

Let A and B be rational numbers such that 4 A3 + 27 B2 6= 0. An ellipticcurve E is the set of all (X , Y ) satisfying the equation

Y 2 = X 3 + A X + B.We will also include the “point at infinity” O.

Example: Y 2 = X 3 − 36 X is an elliptic curve.

Non-Example: Y 2 = X 3 − 3 X + 2 is not an elliptic curve.

-12 -8 -4 0 4 8 12

-20

-10

10

20

-3 -2 -1 0 1 2 3

-5

-2.5

2.5

5





Example: Quartic Curves

Proposition

Fix a rational number k different from −1, 0, 1. Then the quartic curve

E : y2 =(1− x2

) (1− k2 x2

)is an elliptic curve.

Proof: Make the substitutions

x =X − 3 (5− k2)

X − 3 (5 k2 − 1)

y =6 (k2 − 1)Y

(X − 3 (5 k2 − 1))2

⇐⇒

X =

3 (5 k2 − 1) x − 3 (5− k2)

x − 1

Y =54 (k2 − 1) y

(x − 1)2

Then E is birationally equivalent to the curve

Y 2 = X 3 +A X +B where

A = −27(k4 + 14 k2 + 1

)B = −54

(k6 − 33 k4 − 33 k2 + 1

).





Chord-Tangent Method

Fix an elliptic curve E . Given two rational points, we explain how toconstruct more.

1 Start with two rational points P and Q.

2 Draw a line through P and Q.

3 The intersection, denoted by P ∗ Q, is another rational point on E .





Example: Y 2 = X 3 − 36 X

Consider the two rational points

P = (6, 0) and Q = (12, 36).

-8 -4 0 4 8 12 16 20 24

-50

50

P

Q

P*Q

P ∗ Q = (18, 72)







P = (6, 0) and Q = (12, 36).

-8 -4 0 4 8 12 16 20 24

-50

50

P

P ∗ P = OSUMSRI Number Theory Seminar Elliptic Curves: A Survey






P = (6, 0) and Q = (12, 36).

-8 -4 0 4 8 12 16 20 24

-50

50

Q*Q

Q

Q ∗ Q = (25/4, 35/8)





Group Law

Definition

Let E be an elliptic curve, and denote E (Q) as the set of rational pointson E . Define the operation ⊕ as

P ⊕ Q = (P ∗ Q) ∗ O.







P = (6, 0) and Q = (12, 36).

-8 -4 0 4 8 12 16 20 24

-50

50

P

Q

P*Q

P+Q

P ⊕ Q = (18,−72)







P = (6, 0) and Q = (12, 36).

-8 -4 0 4 8 12 16 20 24

-50

50

P

P ⊕ P = OSUMSRI Number Theory Seminar Elliptic Curves: A Survey






P = (6, 0) and Q = (12, 36).

-8 -4 0 4 8 12 16 20 24

-50

50

Q*Q

Q

Q+Q

Q ⊕ Q = (25/4,−35/8)





Poincare’s Theorem

Theorem (Henri Poincare, 1901)

E (Q) is an abelian group under ⊕.

Recall that to be an abelian group, the following axioms must besatisfied:

Closure: If P, Q ∈ E (Q) then P ⊕ Q ∈ E (Q).

Associativity: (P ⊕ Q)⊕ R = P ⊕ (Q ⊕ R).

Commutativity: P ⊕ Q = Q ⊕ P.

Identity: P ⊕O = P for all P.

Inverses: [−1]P = P ∗ O satisfies P ⊕ [−1]P = O.





What types of properties

does this

abelian group have?




Poincare’s Conjecture / Mordell’s TheoremTorsion SubgroupMordell-Weil Rank

Poincare’s Conjecture

Conjecture (Henri Poincare, 1901)

Let E be an elliptic curve. Then E (Q) is finitely generated.

Recall that an abelian group G is said to be finitely generated if thereexists a finite generating set

{a1, a2, . . . , an}

such that, for each given g ∈ G , there are integers m1, m2, . . . , mn suchthat

g = [m1]a1 ◦ [m2]a2 ◦ · · · ◦ [mn]an.

Example: G = Z is a finitely generated abelian group because allintegers are generated by a1 = 1.





Mordell’s Theorem

Theorem (Louis Mordell, 1922)

Let E be an elliptic curve. Then E (Q) is finitely generated.

That is, there exists a finite group E (Q)tors and a nonnegative integer rsuch that

E (Q) ' E (Q)tors × Zr .

The set E (Q) is called the Mordell-Weil group of E .

The finite set E (Q)tors is called the torsion subgroup of E . Itcontains all of the points of finite order, i.e., those P ∈ E (Q) suchthat

[m]P = O for some positive integer m.

The nonnegative integer r is called the Mordell-Weil rank of E .






Consider the three rational points

P1 = (0, 0), P2 = (6, 0), and P3 = (12, 36).

[2]P1 = [2]P2 = O, i.e., both P1 and P2 have order 2. They are torsion.

-8 -4 0 4 8 12 16 20 24

-50

50

(6,0)(-6,0)

(0,0)

E (Q)tors = 〈P1, P2〉 ' Z2 × Z2






Consider the three rational points

P1 = (0, 0), P2 = (6, 0), and P3 = (12, 36).

[2]P3 = (25/4,−35/8) and [3]P3 = (16428/529,−2065932/12167).

-8 0 8 16 24 32

-160

-80

80

160

P

[2]P

[3]P

E (Q) = 〈P1, P2, P3〉 ' Z2 × Z2 × ZSUMSRI Number Theory Seminar Elliptic Curves: A Survey




Classification of Torsion Subgroups

Theorem (Barry Mazur, 1977)

Let E is an elliptic curve, then

E (Q)tors '

{Zn where 1 ≤ n ≤ 10 or n = 12;

Z2 × Z2m where 1 ≤ m ≤ 4.

Remark: Zn denotes the cyclic group of order n.

Example: The elliptic curve Y 2 = X 3 − 36 X has torsion subgroupE (Q)tors ' Z2 × Z2 generated by P1 = (0, 0) and P2 = (6, 0).

Mordell’s Theorem states that

E (Q) ' E (Q)tors × Zr .

What can we say about the Mordell-Weil rank r?





Mordell-Weil Rank

Not much is known about the rank r ...

The highest rank ever found for all known examples of elliptic curvesis r = 28.

Is there a constant B such that r ≤ B for all elliptic curves E ?

Let T be one of the fifteen groups in Mazur’s Theorem, andconsider the collection of all elliptic curves E with E (Q)tors ' T .

Is there a constant B(T ) such that r ≤ B(T )for all elliptic curves with E (Q)tors ' T ?





Records for Prescribed Torsion and RankE(Q)tors Known r ≤ Author (Year)

Z1 28 Elkies (2006)Z2 18 Elkies (2006)Z3 13 Eroshkin (2007, 2008)Z4 12 Elkies (2006)Z5 6 Dujella – Lecacheux (2001)

Z6 8Eroshkin (2008), Dujella – Eroshkin (2008)

Elkies (2008), Dujella (2008)Z7 5 Dujella – Kulesz (2001), Elkies (2006)Z8 6 Elkies (2006)

Z9 3Dujella (2001), MacLeod (2004)

Eroshkin (2006), Eroshkin - Dujella (2007)Z10 4 Dujella (2005), Elkies (2006)Z12 3 Dujella (2001, 2005, 2006), Rathbun (2003, 2006)

Z2 × Z2 14 Elkies (2005)Z2 × Z4 8 Elkies (2005), Eroshkin (2008), Dujella - Eroshkin (2008)Z2 × Z6 6 Elkies (2006)

Z2 × Z8 3Connell (2000), Dujella (2000, 2001, 2006)

Campbell - Goins (2003), Rathbun (2003, 2006)Flores - Jones - Rollick - Weigandt (2007)

http://web.math.hr/~duje/tors/tors.html


http://web.math.hr/~duje/tors/tors.html




How does one compute r?

The current method uses ideas from Mordell’s proof of the PoincareConjecture:

Mordell proved that the quotient group E (Q)/2 E (Q) is alwaysfinite. Note that

E (Q) ' Z2 × Z2m × Zr =⇒ E (Q)

2 E (Q)' Z2

r+2.

It suffices to find a way to countthe number of elements in this finite group.

The only known method for counting the number of elements mighttake days – or even years – to finish.

Is there a better method of determining the rank r?

Determining rank is a hard problem!SUMSRI Number Theory Seminar Elliptic Curves: A Survey




How does this

help answer

the motivating questions?




Rational Triangles Revisited

Can we find infinitely many right triangles (a, b, c) having rational sidesand area A = 6?

Proposition

Let X and Y be rational numbers, and denote the rational numbers

a =X 2 − 36

Y, b =

12 X

Y, and c =

X 2 + 36

Y.

Then

a2 + b2 = c2

1

2a b = 6

if and only if

{Y 2 = X 3 − 36 X .

Example: (X , Y ) = (12, 36) corresponds to (a, b, c) = (3, 4, 5).




Rational Triangles Revisited

The elliptic curve E : Y 2 = X 3 − 36 X has Mordell-Weil group

E (Q) = 〈P1, P2, P3〉 ' Z2 × Z2 × Z

as generated by the rational points

P1 = (0, 0), P2 = (6, 0), and P3 = (12, 36).

P3 is not a torsion element, so we find triangles for each [m]P3:

[1]P3 = (12, 36) =⇒ (a, b, c) = (3, 4, 5)

[−2]P3 =

„25

4,

35

8

«=⇒ (a, b, c) =

„49

70,

1200

70,

1201

70

«[−3]P3 =

„16428

529,

2065932

12167

«=⇒ (a, b, c) =

„7216803

1319901,

2896804

1319901,

7776485

1319901

«

There are infinitely many rational right triangles with area A = 6!




Acknowledgments

We would like to thank...

National Security Agency (NSA)

National Science Foundation (NSF)

Miami University

Edray Goins and Beth Fowler

Dennis Davenport and Vasant Waikar

Other SUMSRI Participants


elliptic curves: a surveyegoins/notes/4-covering...july 17, 2008 sumsri number theory seminar...

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