How do Noetherian modules
work on the Mordell theorem?
(There are some inaccuracies in my abstract)
Chu Chu
Preliminary knowledge
Citation from Rational Points on Elliptic Curves (2015) - Silverman, Tate chapter 1.2
Additive:
P*Q is we draw the line joining P to Q and obtain the third point.
By identity point, O generates more points by addition (P+Q) and draws elliptic curves.
P, Q, and O are not colinear. O, P*Q, and P+Q colinear. It cannot generate a group if P, Q, and O are colinear.
Rational Points on Elliptic Curves (2015) - Silverman, Tate chapter 1.2
Identity:
if we join P to O, then we get the point P ∗ O as the third intersection point. Next, we join P ∗ O
to O and take the third intersection point. That third intersection point is clearly P.
Therefore, P + O = P.
Rational Points on Elliptic Curves (2015) - Silverman, Tate chapter 1.2
Abelian condition:
P + Q = Q + P, since the line through P and Q is the same as the line through Q and P, so P ∗ Q = Q ∗ P.
Inverse:
Draw the tangent line to the cubic at O, and let the tangent meet the cubic at the additional point S, i.e., S = O∗O.
(We are assuming that the cubic is non-singular, so there is a tangent line at every point.) Then given a point Q, we
join Q to S, and the third intersection point Q ∗ S will be −Q. Q + (−Q) = O.
Rational Points on Elliptic Curves (2015) - Silverman, Tate chapter 1.2
Associative law: As a figure, (P+Q)*R=P*(Q+R) because they are on the same point. To add
P +Q to R, we draw the line through P +Q. That meets the curve at (P + Q) ∗ R, so to get (P +
Q) + R, we have to join (P +Q)∗R to O and take the third intersection. Now that does not
show up too well in the picture, but to show (P + Q) + R = P + (Q + R), it will be enough to
show that (P + Q) ∗ R = P ∗ (Q + R).
Rational Points on Elliptic Curves (2015) - Silverman, Tate chapter 1.2
By definition of an additive abelian group,
1)Addition
2)Abelian: P+Q=Q+P
3)inverse: Q+(-Q)=O
4)Identity element: P+O=P
5)associative: (P+Q)+R=P+(Q+R){(P+Q)*R=P*(Q+R) }
We conclude that elliptical curves can be described by additive
abelian groups in general.
Mordell’s theorem says that we get all of the rational points by
starting with a set of points, drawing lines through those points to
get new points, then drawing lines through the new points to get yet
more points, and so on. In terms of the group law, this says that the
group of rational points is finitely generated. Therefore, we have
the following statement of Mordell’s theorem:
If a non-singular rational plane cubic curve has a rational point,
then the group of rational points is finitely generated(Mordell
theorem)
P=O, 2P=O and 3P=O(figure from
https://fangpenlin.com/posts/2019/10/07/ellipticcurve-cryptography-explained/)
Main body
A torsion group is a group in which every element has finite order. In
this paper, they provide examples of 2P = O and 3P=O are torsion
points. To verify that a point P has finite order, one must find an integer
n ≥ 1 such that nP = O. By Mordell theorem we conclude that the
additive abelian group used to describe elliptic curves generated by
nP=O are torsion groups because nP=O are torsion points(Torsion
Points of Elliptic Curves Over Number Fields by Christine Croll
chapter 5, 2006). For n is the order of the group.
There is a strong relationship between torsion and Noetherian.
Noetherian condition satisfied ascending chain condition. Because the
elements of the torsion group are finite, there exists a maximum value
of ascending chain abelian subgroups, which are classified by torsion
points nP=O. For n are positive integers and n less and equal to 12, for
elliptic curves defined over Q (Mazur’s Theorem). But Billing and
Mahler proved in 1940 that not exists curves of order 11. Based on n
being finite, each torsion point has finite order, there exists a maximal
order subgroup. (Zorn’s lemma). It satisfies Noetherian conditions, for
elliptic curves defined over Q.
Zorn’s Lemma: If S is any nonempty partially ordered set in which
every chain has an upper bound, then S has a maximal element.
The groups listed in Mazur’s Theorem are possible finite subgroups,
but the full group itself might very well be infinite.
Rational Points on Elliptic Curves (2015) - Silverman, Tate chapter 2.5
Citation from On exceptional points on cubic curves. (1940)-G. Billing, K. Mahler, J. Lond. Math. Soc. 15,32–43.
Here is a picture directly from the paper.
“”, Rational Points on Elliptic Curves (2015). so 4 lies on the curve if n=11. Any non-singular curve given by a cubic
equation is a curve of genus 1
By definition of modules, modules are additive abelian
groups and scalars multiplication over a ring. An
additive abelian group can be viewed as a module over
the ring Z. The group generated by nP=O and gets more
points by the addition rule means there exists a
maximum order subgroup similar to the Noetherian
module which has a maximum value of ascending chain
means a maximal submodule exists.
Therefore, the abelian groups for elliptic curves defined
over Q are Noetherian modules.
Reference:
1) Rational Points on Elliptic Curves (2015) - Silverman, Tate chapter 1.2 and
chapter 2.5
2) Elliptic Curve Cryptography Explained(2019) – Fang-Pen
https://fangpenlin.com/posts/2019/10/07/elliptic-curve-cryptography-explained/
3) Torsion Points of Elliptic Curves Over Number Fields(2006) - Christine Croll
chapter 5
https://people.math.umass.edu/~hajir/m499d/cc-thesis.pdf
4) On exceptional points on cubic curves. (1940)-G. Billing, K. Mahler, J. Lond.
Math. Soc. 15, 32–43
https://carmamaths.org/resources/mahler/docs/067.pdf