Presentation

Rational Points on Elliptic Curves
Pradeep Kumar Mishra
Department of Mathematics
Indian Institute of Technology Hyderabad
May 4, 2015
Pradeep Kumar Mishra Rational Points on Elliptic Curves

Elliptic curve
Definition
1 An elliptic curve is a pair (E, O), where E is a non singular
cubic curve and O ∈ E in projective space (We generally
denote the elliptic curve by E, the point O being understood).
The elliptic curve E is defined over field K, written E/K, if E
is defined over field K as a curve and O ∈ E.

Elliptic curve
Deﬁnition
2 An aﬃne Weierstrass equation (W.E.) over K is an equation
of the form:
E : Y 2 + a1XY + a3Y = X3 + a2X2 + a4X + a6, ai s ∈ K.

Elliptic curve
Deﬁnition
of the form:
c4 = (a2
1 + 4a2)2 − 24(2a4 + a1a3)
j = c4/ for = R(a,a2, a3, a4, a6) = 0

Elliptic curve
Deﬁnition
of the form:
c4 = (a2
1 + 4a2)2 − 24(2a4 + a1a3)
j = c4/ for = R(a,a2, a3, a4, a6) = 0
is called the discriminant of E, j its j-invariant. A
non-singular aﬃne W.E is an Elliptic curve.
Riemann-Roch theorem says every E.C. can be written as a
Weierstrass plane cubic, and conversely, every non-singular
Weierstrass plane cubic curve is an E.C.

Group law on an elliptic curve
• If char(K) = 2, 3, then by change of variable, E looks like
E : y2
= x3
+ ax + b

E : y2
= x3
+ ax + b
Proposition
(a) The elliptic curve given by a Weierstrass equation is
non singular if and only if = 0.

E : y2
= x3
+ ax + b
Proposition
(b) Two elliptic curves are isomorphic over ﬁeld ¯K if and
only if they both have the same j-invariant.

E : y2
= x3
+ ax + b
Proposition
(b) Two elliptic curves are isomorphic over ﬁeld ¯K if and
only if they both have the same j-invariant.
Group law on rational points on an elliptic curve:

Doubling a Point P on E.

Doubling a Point P on E.
Vertical Lines and an Extra Point
at Inﬁnity.

Algebraic Formulas for Addition on E
Suppose that we want to add
points
P = (x1, y1) and Q = (x2, y2)
on the elliptic curve
y2 = x3 + ax + b, a, b ∈ Q.
If P1 = P2 then λ = y2−y1
x2−x1
and if P1 = P2 then λ =
3x2
1 +a
2y1
Then P + Q = (λ2 − x1 − x2, −λ3 + 2λx1 + λx2 − y1) .

Algebraic Formulas for Addition on E
Suppose that we want to add
points
P = (x1, y1) and Q = (x2, y2)
on the elliptic curve
y2 = x3 + ax + b, a, b ∈ Q.
If P1 = P2 then λ = y2−y1
x2−x1
and if P1 = P2 then λ =
3x2
1 +a
2y1
Then P + Q = (λ2 − x1 − x2, −λ3 + 2λx1 + λx2 − y1) .
Observation:
If a, b ∈ Q and P, Q ∈ E(Q) , then P + Q, 2P ∈ E(Q).

Properties of “Addition” on E
The addition law on E has the following properties:
(a) P + O = O + P = P ∀ P ∈ E(Q)
(b) P + (−P) = O ∀ P ∈ E(Q)
(c) P + Q = Q + P ∀ P, Q ∈ E(Q)
(d) P+(Q+R) = (P+Q)+R ∀ P, Q, R ∈ E(Q)
(e) P +Q ∈ E(Q) ∀ P, Q ∈ E(Q)
In other words, the addition law + makes the points of E
into a commutative group.

Rationality of points in the torsion subgroup

Theorem (Nagell 1935, Lutz 1937)
The ﬁnite order points (x, y) ∈ Q2 on an elliptic curve E satisfy:

1 x and y are integers

1 x and y are integers such that either y = 0 or y| .

2 There is a ﬁnite number of such points.

2 There is a ﬁnite number of such points.
This helps us compute E(Q)tors.
Mazur’s Theorem
If P ∈ E(Q)tors, then 1 ≤ ordE (P) ≤ 12 with ordE (P) = 11.

Computation of E(Q)tors
Torsion and non-torsion element
There is a simple algorithm for computing E(Q)tors.
1 Find integers y where y|D.
2 For every y found above, ﬁnd roots of f (x) − y2 to obtain
x-coordinates.

Computation of E(Q)tors
Torsion and non-torsion element
There is a simple algorithm for computing E(Q)tors.
1 Find integers y where y|D.
2 For every y found above, ﬁnd roots of f (x) − y2 to obtain
x-coordinates.
3 For every P = (x, y), compute nP where n = 2, 3, . . . , 10, 12.
If nP = O then P ∈ E(Q)tors.
If nP has non-integer coordinates,P /∈ E(Q)tors.

The Mordell-Weil Theorem
Let E be an elliptic curve deﬁned over Q.

Let E be an elliptic curve deﬁned over Q. Then E(Q) is
ﬁnitely generated Abelian group.

ﬁnitely generated Abelian group. (Mordell 1922).

E(Q) ∼= Zr ⊕ E(Q)tors

where r is the rank of elliptic curve.

where r is the rank of elliptic curve.
E(Q)tors.
Definition
Height of the point P = (x, y) ∈ E(Q) is defined as :
H(P) = H(x) = H
m
n
= max{|m|, |n|}
further we define
h(P) = logH(P)
So h(P) is always a non negative real number.

Mordell-Weil Theorem
For proving Mordell-Weil Theorem we need following four
lemmas:
Lemma 1 : for every real number M, the set
{P ∈ E(Q) : h(P) ≤ M} is ﬁnite.

lemmas:
Lemma 2 : for every P0 ∈ E(Q), there is a constant κ0
depending on P0, a, b, c so that
h(P + P0) ≤ 2h(P) + κ0 ∀P ∈ E(Q).

lemmas:
h(P + P0) ≤ 2h(P) + κ0 ∀P ∈ E(Q).
Lemma 3 : there is a constant κ so that
h(2P) ≥ 4h(P) − κ ∀P ∈ E(Q).

lemmas:
h(P + P0) ≤ 2h(P) + κ0 ∀P ∈ E(Q).
Lemma 3 : there is a constant κ so that
h(2P) ≥ 4h(P) − κ ∀P ∈ E(Q).
Lemma 4 : The subgroup 2E(Q) has ﬁnite index in E(Q).

Proof of Mordell-Weil Theorem
Let {Qi }n
i=1 be representatives for the cosets. Then there is an
index i1, depending on P, such that
P − Qi1 ∈ 2E(Q) ⇒ P − Qi1 = 2P1 f.s. P1 ∈ E(Q).
Now we do the same thing with P1. Continuing this process, we
ﬁnd we can write
P1 − Qi2 = 2P2, P2 − Qi3 = 2P3, P3 − Qi4 = 2P4, . . .
Pm−1 − Qim = 2Pm
where {Qij }, 1 ≤ i ≤ n, 1 ≤ j ≤ m are chosen from the coset
representatives {Qi }n
i=1

and {Pi }m
i=1 are elements of E(Q).From above we have
P = Qi1 + 2P1
Now substitute the second equation P1 = Qi2 + 2P2 into this to get
P1 = Qi1 + 2Qi2 + 4P2
Continuing in this fashion, we obtain
P = Qi1 + 2Qi2 + 4Qi3 + · · · + 2m−1
Qm + 2m
Pm (a)
From 2nd lemma
h(P − Qi ) ≤ 2h(P) + κi ∀P ∈ E(Q)
We do this for each Qi , 1 ≤ i ≤ n. Let κ be the largest of the κi s.

Then
h(P − Qi ) ≤ 2h(P) + κ ∀P ∈ E(Q) and 1 ≤ i ≤ n
Now from lemma (3)
4h(Pj ) ≤ h(2Pj ) + κ = h(Pj−1 − Qij
) + κ ≤ 2h(Pj−1) + κ + κ
h(Pj ) ≤
3
4
h(Pj−1) −
1
4
(h(Pj−1) − (κ + κ))
From this we see that if h(Pj−1) ≥ κ + κ then
h(Pj ) ≤
3
4
h(Pj−1)

So we will find an index m so that h(Pm) ≤ κ + κ
We have now shown that every element P ∈ E(Q) be written in
the form
P = a1Q1 + a2Q2 + a3Q3 + · · · + anQn + 2m
Pm
for certain integers a1, a2, a3, . . . , an and some point Pm ∈ E(Q)
satisfying the inequality h(Pm) ≤ κ + κ . Hence, the set
{Q1, Q2, Q3, . . . , Qn} ∪ {Pm ∈ E(Q) : h(Pm) ≤ κ + κ }
generates E(Q) moreover this set is finite so E(Q) is finitely
generated Abelian group.

Finding the rank of E(C)
There is no known eﬀective method to ﬁnd r, the rank.
For example the curve y2 = x3 + 877x has rank one and the
x−coordinate of a generator is given by
612776083187947368101
7884153586063900210
2
It is conjectured that there are curves of arbitrarily high rank
As of 2006 the curve of highest known rank is of rank at least
28. It is given by
y2
+xy +y = x3
−x2
−2006776241557552658503+320820933
8542750930230312178956502x + 344816117950305564670
3298569039072037485594435931918036126600829629193
9448732243429

ECC
ECC

Application(congruent number problem)
• A natural number n is called congruent number if there exist a
right angled triangle with all three sides rationals and with area n.
Suppose a, b, c satisﬁes: a2 + b2 = c2 and 1
2ab = n Then
the set
n(a + c)
c
, y =
2n2(a + c)
b2
⇒ y2
= x3
− n2
x
Conversely, a = x2−n2
y , c = x2+n2
y , b = 2nx
y
• A positive rational number n is congruent if and only if the
elliptic curve has a rational point with y = 0.
Continuing with n = 5, y2 = x3 − 25x. We have poiont (−4, 6) on
the curve E
Now we can ﬁnd a, b and c.

Further study
Furture work plan:
1 Can we find an elliptic curve where all points are of infinite
order.
2 Elliptic curves over finite fields.

References
Koblitz, Neal. Introduction to elliptic curves and modular
forms. Second edition. Graduate Texts in Mathematics, 97.
Springer-Verlag, New York, 1993. Graduate Texts in
Mathematics.
Silverman, Joseph H. The arithmetic of elliptic curves. Second
edition. Graduate Texts in Mathematics, 106. Springer,
Dordrecht, 2009.
David Cox, John Little, Donal O Shea, Ideals, Varieties, and
Algorithms,Third edition, 2007.
Silverman, Joseph H.; Tate, John. Rational points on elliptic
curves. Undergraduate Texts in Mathematics. Springer-Verlag,
New York, 1992.
Shafarevich, Igor R. Basic algebraic geometry. 2. Schemes and
complex manifolds. Third edition. Translated from the 2007
third Russian edition by Miles Reid. Springer, Heidelberg, 2013.

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