2015-10-12 김창헌 1 Torsion of elliptic curves over number fields ( 수체 위에서...
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Transcript of 2015-10-12 김창헌 1 Torsion of elliptic curves over number fields ( 수체 위에서...
23年 4月 21日 김창헌 1
Torsion of elliptic curves over number fields ( 수체 위에서 타원곡선의 위수구조 )
발표 : 김창헌김창헌 ( 한양대학교 )전대열 ( 공주대학교 ), Andreas Schweizer (KAIST) 박사와의 공동연구임
23年 4月 21日 김창헌 2
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
23年 4月 21日 김창헌 3
Pythagorean Theorem
Pythagoraslived approx 569-475 B.C.
23年 4月 21日 김창헌 4
Pythagorean Triples
Triples of whole numbers a, b, c such that2 2 2a b c
23年 4月 21日 김창헌 5
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
23年 4月 21日 김창헌 7
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.
23年 4月 21日 김창헌 8
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
23年 4月 21日 김창헌 9
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
23年 4月 21日 김창헌 10
The Secant Process
2 3y y x x
( 1,0) & (0, 1) give (2, 3)
23年 4月 21日 김창헌 11
The Tangent Process
23年 4月 21日 김창헌 12
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.
23年 4月 21日 김창헌 13
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
23年 4月 21日 김창헌 14
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
23年 4月 21日 김창헌 15
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
23年 4月 21日 김창헌 16
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
23年 4月 21日 김창헌 17
(x,y) = (1,0) xx yy
23年 4月 21日 김창헌 18
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.
23年 4月 21日 김창헌 19
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 :
23年 4月 21日 김창헌 44
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 .
23年 4月 21日 김창헌 45
Torsion subgroups
23年 4月 21日 김창헌 46
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 :
23年 4月 21日 김창헌 49
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 .