Video summary
The Mordell-Weil theorem is a foundational result in the arithmetic geometry of elliptic curves, stating that for any elliptic curve defined over the rational numbers $\mathbb{Q}$, the group of its rational points forms a finitely generated abelian group. This means that while there can be infinitely many rational solutions to equations like $y^2 = x^3 - 2$, these points are not scattered randomly but instead form an algebraic structure where every point can be expressed as a finite combination of a specific set of generator points using the curve's unique addition law. The theorem was originally proven by Louis Mordell for elliptic curves over $\mathbb{Q}$ and later generalized by André Weil to abelian varieties, while Serge Lang extended its applicability to algebraic number fields beyond just the rationals.
The proof strategy relies on two key properties: a weak version of the theorem regarding multiplication maps and the concept of height functions. The weak Mordell-Weil theorem asserts that if one takes all rational points on an elliptic curve $E$ and divides them by 2 (forming the quotient group $E(\mathbb{Q})/2E(\mathbb{Q})$), the resulting set is finite. To bridge this to the full theorem, mathematicians utilize a height function, which measures the complexity of a point based on its projective coordinates; specifically, doubling a point roughly quadruples its height. By showing that there are only finitely many points with bounded height and using induction, one can demonstrate that any arbitrary rational point must be generated by those few low-height representatives, thereby proving the group is indeed finitely generated.
While the theoretical proof establishes finite generation effectively in most practical scenarios, a significant obstacle remains when attempting to make the algorithm for finding these generators completely explicit or "effective." The primary difficulty lies with the Tate-Shafarevich group, which acts as an obstruction that measures the failure of local solubility conditions to imply global solutions. Although this group is conjectured to be finite in all cases and appears so in every calculated instance, there is currently no rigorous proof guaranteeing its finiteness for every possible elliptic curve. If a counterexample were found where this group was infinite, it would fundamentally alter our understanding of the arithmetic properties of these curves; however, as things stand, we can effectively compute the finite-order points but cannot definitively prove that an algorithm exists to find all generators without relying on the unproven finiteness of the Tate-Shafarevich group.
Read the full video transcript
this lecture will be about the
Modell Bay theorem so I will start by
recalling what this theorem says so um
Mell more than a century ago show that
if we've got an elliptic
curve um let's call it e over the
rational numbers Q then the group of
points on the elliptic curve with
coordinates dered by E of Q is finitely
generated as a group so I just quickly
remind um recall what this means so an
elliptic Curve will look something like
y^2 = X Cub - 2 and we can ask um what
are the rational Solutions of this for
example it's got a rational solution of
Y = 5 and x =
3 um and the number of rational
Solutions may be infinite but Modell
showed that the they form a group and
this group is actually finitely
generated so um just recall what this
group structure looks like if we've got
an elliptic curve it might look
something like
this and if we've got three points on
the elliptic curve with rational
coordinates then we say that there sum
is equal to zero if they lie on a line
straight line so here are the points a b
and c and in this case their sum is zero
so I'd better make a couple of comments
about this first of all this sum has
nothing to do with adding up their
coordinates this is defining um a group
operation of addition on the points of
an elliptic curve um secondly this
doesn't quite Define the group structure
because I haven't said what the origin
of the group is and you can just pick
any point on the elliptic curve to be
the the origin of a group for example
you could take the point at
Infinity um so Mell proved this finite
generation for points of an elliptic
curve over the rational numbers um this
was generalized by Andre V who um
changed elliptic curves to a billion
varieties um and it was extended further
by Lang who um showed that you could
change Q to any algebraic number
field um there are also versions of this
over rational function Fields but you
have to be a little bit more careful
about
that
um so the proof of the M delve
theorem um basically follows Modell's
original argument with some various
improvements although Mell would
probably not have considered them
improvements so um first will we prove
the weak mod delv
theorem so what for this you take the
group of points over the elliptic curve
and multiply them all by two using the
group law on the elliptic curve and we
get a group E of Q over 2 e of Q as the
quotient and the weak model V theorem
says that this group here is
finite so if e of Q is finite to
generate this group obviously has to be
finite and the converse is not doesn't
automatically follow for example if you
take the group of rational numbers and
multiply things by two then you get Q
over 2 q and this is certainly finite
because it's just the the trivial group
but the rational numbers are not
finitely generated so the weak more Del
V theem by itself is is not
enough and to get the full mod Del V
theorem we also need the height
a point on elliptic curve and what this
says is that e of Q has a
symmetric um bilinear
form um and this is the following
property that um it's more or less
positive definit so a a is greater than
zero if a um is not a finite
order and the key property is that the
that there are only a finite number of
points on the elliptic curve with
rational coordinates with um
any um with with bounded with with value
of a bounded by some fixed constant so
if you fix any constant M like a million
they're only a finite number of points
um with lengths at most that where of
course the length of a point is just the
square root of its um the inner product
with itself
um so um the first thing to do is to
check that properties one and two imply
the full Model A theem so we're going to
show that properties 1 + 2 implies more
Del
V and for this what we do is we pick a
set of points A1 to a n
representing all the points of e of
Q over 2 e of Q so these points here are
IR rational points on the elliptic curve
and we pick enough of them so that um
every Point here is represented by one
of them which we can do because we
recall this group is
finite and now we pick
M so um all these points a I have length
less than
M and then we're going to show that e of
Q is
generated by the points a of of length
um um at most
M um and this argument is very easy
suppose we've got any point then we can
write um this point x as 2y plus some
AI um this is just saying that um the AI
are a set of Representatives for the the
curve modulo 2 times the curve and now
we notice that if
Y is length greater than M then X has
length greater than y because a has
length at most M and 2 Y is is is going
to
have uh is going to have length at most
the length of X plus the length of AI
and we can rewrite this as saying that Y
is less than x and
less X um has length at most
M um and this shows that every point x
can be written as a linear combination
of points of length at most M and you do
this by induction you keep rewriting X
as 2 y + a i with y less than x and you
can keep doing this until X is length at
most M so you can reduce every point to
points of length at most
M so this shows that the um full delum
follows from the the two properties I
mentioned um now we discuss what what is
the height of a
point so for an elliptic
curve um you can embed it in the
projective plane and then it's got
projective coordinates X Y and Z and if
the point is rational coordinates then
we can assume x y and z are integers and
we can also take them be Co
Prime and then we can Define the
logarithmic height to be the logarithm
of the maximum of the absolute values of
X Y and
Z and this isn't quite the height we we
really want so H is
approximately
quadratic um so in particular h of 2x is
approximately 4 h of
X um and um you can actually work with
this approximately quadratic function in
fact which is what Modell originally did
it makes the argument a little bit
fussier because you need to keep track
of what the error terms um for this
being approximately quadratic is um take
notice that you can make it exactly
quadratic by sort of averaging it um you
define H hat of x to be just the limit
as n tends to Infinity of H of
NX
over n s and then um h of X just turns
out to be
um um the inner product of X with itself
it's suitable by linear
form um for Aon varieties this is rather
similar um what you've got to do is
embed the ailan variety a inside
projector space um for some n um and if
you've done algebraic geometry you know
that to embed something into projector
space you need to choose a line bundle
on
it and line bundles um give you lots of
projective embedding so you choose a
line bundle on your aan variety you
probably want this line bundle to be
symmetric um and this gives you a
bilinear form and if L is
ample whatever that means this turns out
to imply the the form is um positive
definite except on torsion points so
this is essentially Andre V's
contribution he he showed how to how to
extend the height to
um um more General a bilan
varieties um
so um for elliptic curves
um the light there's there's basically a
smallest possible ample line bundle so
there's no real real Choice involved
here you you essentially just put your
electric curve into projector space um
for a billion varieties you get quite a
lot of different heights depending on
which line bundle you
choose um next we want to discuss the
weak mod Del V theorem um where you take
an elliptic curve and you map it to an
elliptic curve by multiplying it by two
and look at the
quotient um in fact more generally we
can work with an isogeny from the
elliptic curve with itself from one
elliptic curve to another elliptic curve
so suppose A and B are elliptic curves
or for that matter a billion varieties
we choose a map from one to the other
which is an isogeny that that means
roughly it has to be
subjective and um has a
kernel and we want this to be
finite so isogeny roughly means that the
kernel is finite and and the map is
subjective
um um so for example if we multiply by
any positive integer that would be an
isogeny and there are also some other
isogenies related to complex
multiplication which we don't need to
worry about and what we want to show is
that b of Q over Lambda a of Q is
finite and for this what we do is we
take K to be the field over Q
generated by the
coordinates of the points um
x with Lambda X
um
um being a rational point of B and we
find we get an exact sequence zero goes
to the kernel of Lambda goes to a of
K goes to the image of a of
K goes to zero where this is contained
in B of
K and what we can do is we can act on
this sequence by the group G which is
just the gwa group of K Over q and here
we've got an exact sequence and if we
take fixed points under this exact
sequence we get Z
goes to the kernel of Lambda over the
rational points of this which is a
finite group of very small order and
this maps to a of Q which maps to under
Lambda to B of Q and this map isn't
surjective um taking fixed points under
a group action is only left exact um
instead um this maps to a certain
cohomology group which is H1 of G the
values incur Lambda which we won't worry
about too
much but what we notice is that b of Q
over Lambda a of
Q
is um a subgroup of this group
here so um what we want to do is to show
that this group is
finite um well the kernel of Lambda is
certain
finite and a first cohomology group of G
with coefficients in something as finite
um is finite if G is finite so we want
to know is G
finite in other words um is the degree
of this field extension K Over Q finite
and if we um do that then we've finished
off for proof of a weak more delve
theorem so I'll explain how to do that
so what we've done is we've reduced our
problem to the following so K is
generated by the coordinates of points
x with Lambda X having rational
coordinates and these
coordinates lie in
fields with the following properties
first of all the
degree is is
bounded you can bound it in terms of the
degree of Lambda secondly you can check
that it's unramified
outside a finite set of
primes um so um what you do is you've
got a few bad primes and the bad primes
come from the following so first of all
we we we take a model for um um A and B
over over the Ring of integers of some
number field and ask for which primes do
they not
reduce do they become singular curves
when you reduce modu of that Prime so
these would be points of bad
reduction and there're a finite number
of those and then we can um
primes dividing
the degree of Lambda and then we want
primes to
make various Rings into principal ideal
domains so you know if you've got the
Ring of integers of an algebraic number
field it's not usually a principal ideal
domain but you can make it into a
principal ideal domain just by
localizing at a few primes so you add in
all these and this gives you a finite
collection of bad primes and you
probably throw in the prime two as well
because the prime two always goes
wrong um and thirdly these extensions
are a bilan in fact the galwa group
turns out to be a a subgroup of the
additive group of the elliptic
curve and if you take these conditions
along to your friendly neighborhood
algebraic number theorist they will get
very excited and um give you a long
explanation involving class field theory
of how they're only a finite number of
fields with these properties um in fact
they will list them all for you in fact
you don't really need class field field
Theory um 1 2 and three um imply a
finite number of fields which you can
just do using comma
Theory um what you're doing is you're
looking at Bon
extensions and you can get all the Bon
Extensions by first of all throwing a
lot of roots of unity and then taking
nth roots of various elements of your
field and the the the number of roots
you take n is bounded because the degree
is bounded
and the the number of possible a um is
is reasonably bounded because um the
field has to be unramified outside some
set of primes so a can only be divided
by a finite set of primes and this this
means they're only a finite number of
integers a and then you've got to fuss
around the roots of unity of the field
being finitely generated but that that's
a that's a well-known theorem so um
proving there finite number of fields
with these three properties is is a a
routine piece of algebraic number Theory
so that shows that this this field K is
a finite degree so the the Galla group
is finite and so that funny um
cohomology group is finite so that
completes the sketch of the proof of the
M delve
theorem um well there's one very
annoying problem that this is actually
not effective
it doesn't actually
give an algorithm to find the points
well actually it probably does give an
algorithm but rather annoyingly we can't
actually prove that this algorithm
always
works um so what is
effective well things we definitely can
calculate I mean we definitely have an
algorithm for um first of all the height
is completely effective the set of bad
primes s we can list explicitly we can
find the field K Over Q quite explicitly
and we can calculate the group H1 of
galwa K Over Q um with coefficients and
kernel of Lambda um um explicitly it say
it's not very difficult to calculate
it's definition looked like a bit of a
mess but in practice it's quite easy to
handle um um we think of this as being a
known fairly easy group um the problem
arises when we look at a q over Lambda B
of
Q and we had an injective map from a of
Q into this group here and um the
problem is what is the
image and
um well what's the problem you say well
if something is in the image it's
actually quite easy to show it's in the
image because you can just calculate
until you find a point mapping to it the
problem is if a point of this group
isn't in the image of this group then
it's not clear if you've got an
algorithm to solve this problem um the
obstruction to this is a group called
the tatea farich group um which is um
notoriously difficult to calculate it's
believed to be
finite um or conjectured to be finite
and seems to be finite in all cases
we've been able to calculate it and if
it's finite then we can calculate the
image of this
group in an effective way if it turns
out to be infinite for some elliptic
curve then we'd be kind of stuck and
actually this would be really
interesting and really annoying and
people would have to completely rethink
everything they know about elliptic
curves um so the Tate cha farich group
is at the moment the problem to making
this um effective I should say in
practice we can always effectively find
the um group of points of finite order
and elliptic curve we just we're just
not certain that this algorithm always
works