The Modularity Theorem " Every
Elliptic Curve is Modular "
(2017-07-06) The Proof of Fermat's Last Theorem (FLT)
Ken Ribet showed that the modularity conjecture implies FLT.
The story which utimately led to a proof of Fermat's Last Theorem
arguably starts with a 1954 theorem by
Martin Eichler (1912-1992)
which was the first multivariate (a.k.a. indeterminate)
and also the first example of a non-abelian reciprocity law.
It was clear that some elliptic curves are
associated with a modular form, but
the Taniyama–Shimura–Weil conjecture (or modularity conjecture)
stated that such was the case for all of them.
Since its proof in 2001, that's known as the
Modularity Theorem and it can be stated very concisely:
Every elliptic curve is modular.
In 1982, Gerhard Frey (b.1944)
tied the conjecture to FLT by considering an hypothetical counterexample, for an odd prime p :
a p + b p = c p
Frey suspected that the following elliptic curve (the Frey curve)
wouldn't be modular. (Thus making FLT follow from the modularity conjecture.)
y2 = x (x - a p ) (x + b p )
In 1985, Jean-Pierre Serre (b.1926)
analyzed the situation and showed that the validity of Frey's reduction of FLT to the modularity conjecture
only had what seemed to be a tiny gap (Serre called it epsilon and that's how the puzzle
became known for a while).
In 1986, Ken Ribet (b.1948)
found a restricted proof of that missing bit, which he was soon able to generalize with some help from
Barry Mazur (b.1937).
From that point on, a proof of the modularity conjecture would also constitute
de facto a proof of FLT...
(2017-08-01) Elliptic Functions: Lattices and Tori
A torus is like a single cell from a doubly-periodic planar lattice.
By definition, a function of a complex
variable is said to be elliptic if it has two linearly independent periods
(two complex numbers are linearly independent if their real and imaginary parts are not in the same ratio).
(2017-07-06) Elliptic Curves
The name denotes a planar curve, its equation or the underlying group.
An elliptic curve is a
smooth cubic in the projective plane. Let's explain:
The planar curves
whose cartesian equations are polynomial equations in the coordinates (x,y)
are called algebraic curves.
The degree of such a curve is the degree of the polynomial.
Straight lines are algebraic curves of degree 1.
The curves of degree 2 are called conics,
of which there are three types:
ellipses (including circles).
Next up are the algebraic curves of degree 3, which are called
The cubics which don't have any singular points are called elliptic curves.
One example of a cubic which doesn't qualify as an elliptic curve is the
folium of Descartes, because it has a node (i.e., a double point).
Likewise, the curve y2 = x3
isn't an elliptic curve because it has a cusp at the origin.
Through a rational affine change of variables, the equation of an elliptic curve can be
the following canonical form (Weierstrass form):
y 2 = x 3 + a x + b
The idea for the next step comes from an ancient source:
Diophantus first remarked that not
all quadratic equations in two variables have a rational solution.
One example of a quadratic equation without rational solutions is:
x 2 + y 2 = 3
However, if a quadratic equation is known to have at least one
rational solution, then it has infinitely many.
Diophantus obtained all of them by what he called the secant line method
(looking for the unknowns in
terms of displacements from the known solution). This gives all solutions in term
of a single rational parameter (e.g., the ratio of the aforementioned displacements).
For cubic equations, Diophantus couldn't find a similar parametrization
but he made a brilliant remark:
If a cubic equation has two rational solutions,
then a third rational solution is easy to obtain!