Finding a rational point on the elliptic curve y 2 = x 3 +7823.

Finding a rational point on the elliptic curve y2 = x3+7823

Introduction

Mordell-Weil theorem

The group of rational points on an elliptic curve over a number field is finitely generated

So E/Q is finitely generated

How to find the generators?

Mordell curve

An elliptic curve of the form y2 = x3 + D

Generators for |D| 10,000: http://diana.math.uni-sb.de/~simath/MORDELL

All |D| 10,000 except for…

The Mordell-Weil generator

…except for D=7823

Stoll, January 2002: found Mordell-Weil generator via 4-descents

Coordinates of generator:

y2 = x3+7823

Points of finite order on 7823?

Nagell-Lutz: integer coordinates

Calculate discriminant

Check factors

7823: Points of finite order?

Answer: no!

So Mordell-Weil generator has infinite order

7823: Points of infinite order…

Kolyvagin, Gross, Zagier: L-series of E has simple zero at s = 1

So E(Q) is isomorphic to Z and E is of rank 1

One rational point of infinite order generating the Mordell-Weil group

Descent

Descent: C D E

Idea: associate other objects (“covering spaces”) to E

Points on these spaces correspond to points on E via polynomial mapping

Goal: find rational points on these spaces

Descents

Descent via isogeny: “first descent” Isogeny? “2-isogenies”: The curves:

EEEE

babaabxaxyE

babaabxaxyE

baxxyE

4,2,:

4,2,:

:

232

232

32

Descent via 2-isogeny (algorithm)

Given an elliptic curve with point of order 2, transform it to the form E : y2 = x3+ax2+bx, sending point of order 2 to (0,0)

For each square-free divisor d1 of D, look at the homogeneous space and find integer points (M, N, e) which correspond to points

Repeat with Result: E(Q)/2E(Q)

E

4

1

2241

2:1

ed

DeaMMdNCd

31

2

21 ,

e

MNdy

e

Mdx

2-descent

If we can’t find a rational point on some , what went wrong?

(1) Either it has points, but they’re too large to be found in the search

(2) Or has no rational points Carry out 2-descent to distinguish between two

possibilities

1dC

Problem!

But there’s a fundamental problem with the descent via 2-isogeny for the 7823 curve

y2 = x3+7823 is conspicuously lacking a point of order 2

So what do we do if there’s no point of order 2?

Answer: general 2-descent

General 2-descent

General 2-descent (algorithm)

Determine the invariants (I, J) Find the quartics with the given I, J Test equivalence of quartics Find roots of quartics Local and global solubility Recover points on E

General 2-descent

Basic idea is to associate to E a collection of 2-covering homogeneous spaces

a, b, c, d, e are in Q and are such that certain invariants I and J satisfy

I and J are related to certain invariants of E,

for λQ*

edxcxbxaxxgy 2342 )(

66

44 2, cJcI

3222 22727972,312 cebadbcdaceJcbdaeI


Determine the invariants (I, J)

with mwrank:Enter curve: [0,0,0,0,7823]Curve [0,0,0,0,7823] Two (I,J) pairsI=0, J=-211221I = 0, J = -13518144


Find the quartics with the given I, J

d: substitute into original equation



disc=-44614310841Looking for quartics with I = 0, J = -211221Looking for Type 3 quartics:Trying positive a from 1 up to 17Trying negative a from -1 down to -11Finished looking for Type 3 quartics.



Looking for quartics with I = 0, J = -13518144Looking for Type 3 quartics:Trying positive a from 1 up to 68(30,-12,48,116,-18) (41,-16,-6,112,-11) Trying negative a from -1 down to -45(-11,-20,408,1784,2072) (-18,-28,312,996,838) Finished looking for Type 3 quartics.


Test equivalence of quarticsequivalence relation: g1 ~ g2 if

for rational constants

(30,-12,48,116,-18) new (B) #1

(41,-16,-6,112,-11) equivalent to (B) #1

(-11,-20,408,1784,2072) equivalent to (B) #1

(-18,-28,312,996,838) equivalent to (B) #1


Our 2-covering space:

Local and global solubility?

mwrank: locally soluble Global solubility? Local doesn’t imply global,

so we have to search for points… Recovering points on E?

Local (-to-global) solubility

Local solubility?

Given an equation: Check solutions over Qp for all p Check solutions over R

Global solubility?

Checking for a solution over Q if local solubility is known

Local does not imply global (“failure of Hasse principle”)

Failure of Hasse principle

Examples, continued

Examples, continued

Local-to-global article

“On the Passage from Local to Global in Number Theory” (B. Mazur)

http://www.ams.org/bull/pre-1996-data/199329-1/mazur.pdf

Checking local solubility

Back to original problem: we have

We want to check solubility over p-adics and reals

For reals, this is just solving a quartic

For p-adics, situation is a little more complicated

edxcxbxaxxgy 2342 )(

P-adics?

P-adics, continued

P-adics, continued

P-adics, continued

Global solubility?

If locally soluble, then check global solubility

Check up to a certain height on a homogeneous space

If not sure about existence of a rational point, take another descent


Our 2-covering space:

Local and global solubility?

mwrank: locally soluble Global solubility? Local doesn’t imply global,

so we have to search for points… Recovering points on E?

4-descent

4-descent

4-coverings: intersection of two quadric surfaces

Represent these by matrices M1 and M2

Det(xM1 + M2) = g(x)

Stoll

Found matrices with smaller entries with same property

Do this by making substitutions: elements with sums

Swap matrices; repeat

Apply generators of SL4(Z); if made smaller, repeat

Stoll

Found matrices with smaller entries with same property

Do this by making substitutions: elements with sums

Swap matrices; repeat

Apply generators of SL4(Z); if made smaller, repeat

Simultaneous equations:

This has the point (-681 : 116 : 125 : -142)

Which gets us on the 2-covering

Resulting in

on the elliptic curve

4-covering, descent

Calculating “matrixes” [sic]1

1via Nick Rozenblyum

Matrices?

Det(xM1 + M2) = g(x)

The M’s are 4 x 4, symmetric: so 20 unknowns

g is a quartic

Summing, det-ing, equating coefficients…

Laskdfj

lkjsd

a142a232 2a13a14a23a24 a132a242 a142 a22a33

2a12a14a24a33 a11a242 a33 2a13a14a22a34

2a12a14a23a34 2a12a13a24a34 2a11a23a24a34

a122a342 a11a22a342 a132 a22a44 2a12a13a23a44

a11a232a44 a122a33a44 a11a22a33a44 a242 a33b11x

2a23a24a34b11x a22a342 b11x a232 a44b11x

a22a33a44b11x 2a14a24a33b12x 2a14a23a34b12x

2a13a24a34b12x 2a12a342 b12x 2a13a23a44b12x

2a12a33a44b12x 2a14a23a24b13x 2a13a242b13x

2a14a22a34b13x 2a12a24a34b13x 2a13a22a44b13x

2a12a23a44b13x 2a14a232 b14x 2a13a23a24b14x


2a12a23a34b14x a142 a33b22x 2a13a14a34b22x

a11a342b22x a132a44b22x a11a33a44b22x

2a142a23b23x 2a13a14a24b23x 2a12a14a34b23x


2a13a14a23b24x 2a132 a24b24x 2a12a14a33b24x


a142a22b33x 2a12a14a24b33x a11a242 b33x

a122a44b33x a11a22a44b33x 2a13a14a22b34x


2a122a34b34x 2a11a22a34b34x a132 a22b44x

2a12a13a23b44x a11a232 b44x a122 a33b44x

a11a22a33b44x a342 b122x2 a33a44b122 x2

2a24a34b12b13x2 2a23a44b12b13x2 a242b132 x2

a22a44b132x2 2a24a33b12b14x2 2a23a34b12b14x2

2a23a24b13b14x2 2a22a34b13b14x2 a232b142 x2

a22a33b142x2 a342 b11b22x2 a33a44b11b22x2

2a14a34b13b22x2 2a13a44b13b22x2 2a14a33b14b22x2




2a12a34b14b23x2 a142b232 x2 a11a44b232x2




2a13a14b23b24x2 2a11a34b23b24x2 a132b242 x2

a11a33b242x2 a242 b11b33x2 a22a44b11b33x2


2a12a24b14b33x2 a142b22b33x2 a11a44b22b33x2






2a12a13b24b34x2 2a11a23b24b34x2 a122b342 x2

a11a22b342x2 a232 b11b44x2 a22a33b11b44x2


2a12a23b13b44x2 a132b22b44x2 a11a33b22b44x2

2a12a13b23b44x2 2a11a23b23b44x2 a122b33b44x2

a11a22b33b44x2 a44b132b22x3 2a34b13b14b22x3

a33b142b22x3 2a44b12b13b23x3 2a34b12b14b23x3

2a24b13b14b23x3 2a23b142b23x3 a44b11b232 x3

2a14b14b232x3 2a34b12b13b24x3 2a24b132 b24x3

2a33b12b14b24x3 2a23b13b14b24x3 2a34b11b23b24x3

2a14b13b23b24x3 2a13b14b23b24x3 a33b11b242x3

2a13b13b242x3 a44b122 b33x3 2a24b12b14b33x3

a22b142b33x3 a44b11b22b33x3 2a14b14b22b33x3

2a24b11b24b33x3 2a14b12b24b33x3

2a12b14b24b33x3 a11b242b33x3 2a34b122 b34x3





2a11b23b24b34x3 a22b11b342x3 2a12b12b342 x3

a11b22b342x3 a33b122 b44x3 2a23b12b13b44x3

a22b132b44x3 a33b11b22b44x3 2a13b13b22b44x3


a11b232b44x3 a22b11b33b44x3 2a12b12b33b44x3

a11b22b33b44x3 b142b232 x4 2b13b14b23b24x4

b132b242 x4 b142b22b33x4 2b12b14b24b33x4

b11b242b33x4 2b13b14b22b34x4 2b12b14b23b34x4

2b12b13b24b34x4 2b11b23b24b34x4 b122b342 x4

b11b22b342x4 b132 b22b44x4 2b12b13b23b44x4

b11b232b44x4 b122 b33b44x4 b11b22b33b44x4

THE END

Finding a rational point on the elliptic curve y 2 = x 3 +7823.

Documents

descent slide

e slide

e points

introduction slide

eq2eq slide

infinite order slide

descent algorithm

descent isogeny