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Torsion of elliptic curves over number fields ( 수체 위에서 타원곡선의 위수구조 )

발표 : 김창헌김창헌 ( 한양대학교 )전대열 ( 공주대학교 ), Andreas Schweizer (KAIST) 박사와의 공동연구임

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Diophantine equation The main object

of arithmetic geometry: finding all the solutions of Diophantine equations

Examples: Find all rational numbers X and Y such that 122 YX

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Pythagorean Theorem

Pythagoraslived approx 569-475 B.C.

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Pythagorean Triples

Triples of whole numbers a, b, c such that2 2 2a b c

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Enumerating Pythagorean Triples

ax

c

by

c

2 2 1x y Circle of radius 1

Line of Slope t

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If then

is a Pythagorean triple.

Enumerating Pythagorean Triples

rt

s

2 2a s r 2b rs 2 2c s r

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Quadratic equations with rational coefficients Why does the secant method works?

We have a solution Any straight line cuts the circle in 0,1 or 2

points Fact: If we have a quadratic equation

with rational coefficients and we know one solution, then there are infinite number of solutions and they can be parametrized in terms of one parameter.

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what happens with the cubic equations?

Claude Gasper Bachet de Méziriac (1581-1638) :

Let c be a rational number. Suppose that (x,y) is a rational solution of Y2 = X3+c.  Then

is also a rational solution.

),( 3

236

2

4

8820

48

yccxx

ycxx

Bachet

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Cubic Equations & Elliptic Curves

Cubic algebraic equations in two unknowns x and y.

A great bookon elliptic

curves by Joe Silverman

3 33 4 5 0x y

2 3y x ax b

3 3 1x y

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The Secant Process

2 3y y x x

( 1,0) & (0, 1) give (2, 3)

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The Tangent Process

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Elliptic curves Consider a non-singular elliptic curve Y2

= X3+aX2+bX+c Suppose we know a rational solution

(x,y). Compute the tangent line of the curve at

this point. Compute the intersection with the curve. The point you obtain is also a rational

solution.

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Rational points on elliptic curves

Formula: If (x,y) is a rational solution, then (x,y) is another rational solution, where

x = 

it seems that we have found a procedure to compute infinitely many solutions if we know one. But this is not true!

x42bx28cx+b24ac4y2

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Torsion points x =

Problem: If y = 0, x is not defined (or better,

it is equal to infinite). If x = x, and y = y, we get no new

point. What else could happen?

x42bx28cx+b24ac4y2

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Torsion points Beppo Levi (1875-1961)

conjectured in 1908 that there is only a finite number of possibilities, and gave the exact list.Beppo Levi

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Torsion points B. Mazur proved this conjecture

in 1977 in a cellebrated paper. Theorem (Mazur) Let (x,y) be a

rational point in an elliptic curve. Compute x, x, x and x. If you can do it, and all of them are different, then the formula before gives you infinitely many different points.

Barry Mazur

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(x,y) = (1,0) xx yy

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Torsion points In modern language Mazur’s

Theorem says: If (x,y) is a rational torsion point of order N in an elliptic curve over Q, then N <= 12 and N is not equal to 11.

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Mordell’s Theorem

The rational solutions of a cubic equation are all obtainable from a finite number of solutions, using a combination of the secant and tangent processes.

1888-1972

Mordell-Weil group

2 3yyxx

}{}|),{()( 322 baxxyKyxKE

(Mordell-Weil group)

fielda:,,: 32 KbabaxxyE

Mordell-Weil Theorem

Mordell(1888-1972)

K: number field,

The Mordell-Weil group E(K)

is finitely generated.

frtors KEKEKE )()()(

Weil(1906-1998)

E(K)tors: torsion subgroup of

E over K.

Mazur’s Theorem

There are 15 group structures of Etors(Q)of elliptic curves

y2 = x3 + ax + b

for any two rational a and b.

Mazur’s Theorem

The curve X1(N) is of genus 0 iff N = 1–10,12.

Modular curves

• The curve X1(N) is a parametrization of the elliptic curves with a torsion point of order N.

Modular curves

• Tate normal form

.,;)1(:),( 232 KcbbxxbyxycycbE

• E(b,c) satisfies the following:

- P = (0,0): K-rational point,

- ord(P) ≠ 2,3.

(b,c) satisfies FN(b,c) = 0 if and only if

E(b,c) is an elliptic curve with a torsion

point P = (0,0) of order N.

Modular curves

• Modular curve X1(N)

FN(b,c) = 0: the formula arising from

the condition NP = 0.

X1(N): FN(b,c) = 0.

Modular curves)0,0(P

),(2 bcbP

),(3 cbcP

b/cddcdddP ;))1(),1((4 2

)1/();)1(),1((5 22 dceeededeP

e

edg

e

defgfffgP

1

,1

)1());12(,(6 2

Modular curves),0( bP

)0,(2 bP

),(3 2ccP

b/cdddddP ;))1(),1((4 2

)1/());(),1((5 2 dceeddeedeP

Modular curves

5)(ord P•Modular curve X1(5)

: the equation of a

projective line, i.e., X1(5) is of genus 0.

0:)5(1 cbX

)0,(),( bcbc 0 cb

PP 23

X1(11) : y2 + y = x3 – x2 is an elliptic curve,

i.e., X1(11) is of genus 1.

Modular curves

11)(ord P

• Modular curve X1(11)

))(),1(())12(,( 22 eddeedegfffg

)1,1(232 x

yeydxxyy

PP 56

034 322 edededed

Genus table of modular curvesN g1(N) N g1(N)

1 0 11 1

2 0 12 0

3 0 13 2

4 0 14 1

5 0 15 1

6 0 16 2

7 0 17 5

8 0 18 2

9 0 19 7

10 0 20 3

Mazur’s Theorem

The curve X1(N) is of genus 0 iff N = 1-10, 12.

Infinitely many rational points

• X1(N) contains infinitely many rational points if N = 1–10, 12.

• There exist infinitely many elliptic curves defined over Q with rational torsion points of order N for N = 1–10, 12.

Infinitely many rational points

• When does a modular curve has infinitely many K-rational points with a number field K?

• ⇒ E(b,b) is an elliptic curve defined over Q with a rational torsion point of order 5.

0:)5(1 cbX

.;)1(:),( 232 Q bbxxbyxybybbE

Infinitely many rational points• (Mordell-Faltings) Any smooth projective

curve of genus g > 1 defined over a number field K contains only finitely many K-rational points.

• When does a modular curve has infinitely many K-rational points with number fields K of a fixed order?

Kamienny, MazurK : quadratic number fields

– X1(N): of genus 0(rational) iff N = 1–10, 12.

– X1(N): of genus 1(elliptic) iff N = 11, 14, 15.

– X1(N): hyperelliptic iff N = 13, 16, 18.

Kamienny, Mazur

Each of these groups occurs infinitely often as .

There exist infinitely many K-rational points of X1(N) defined over quadratic number fields K for N=1-16,18.

Infinitely many rational points• If there exist a map f : X → P1 of degree d,

then X is called d-gonal.

• If X is 2-gonal and g(X) > 1, then X is called hyperelliptic.

Infinitely many rational points• (Mestre) X1(N) is hyperelliptic for N = 13,

16, 18.

Infinitely many rational points• (Jeon-Kim-Schweizer) X1(N) is 3-gonal iff

N = 1–16, 18, 20 iff

is infinite.

}3]:)([|)({ 1 QQ PNXP

Jeon, Kim, SchweizerK : cubic number fieldsThe group structure that occurs infinitely often as :

Infinitely many rational points• (Jeon-Kim-Park) X1(N) is 4-gonal iff

N = 1–18, 20, 21, 22, 24 iff

is infinite.

}4]:)([|)({ 1 QQ PNXP

Jeon, Kim, ParkK : quartic number fieldsThe group structure that occurs infinitely often as :

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Further Studies Theorem (1996, L. Merel) For

any integer d 1, there is a constant Bd such that for any field K of degree d over Q and any elliptic curve over K with a torsion point of order N, one has that N <= Bd  .

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Torsion subgroups

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Torsion subgroups

Jeon, Kim, SchweizerK : cubic number fieldsThe group structure that occurs infinitely often as :

Jeon, Kim, ParkK : quartic number fieldsThe group structure that occurs infinitely often as :

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Further Studies If d=1, then Bd=12.

If d=2, then Bd=18.

If d=3, then Bd=20?

If d=4, then Bd=24?

감사합니다 .