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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Linking Elliptic Curves & Modular Forms

Cuius rei demonstrationem mirabilem sane detexi. 
Hanc marginis exiguitas non caperet.  

Pierre de Fermat  (1601-1665)   
  • Fermat's Last Theorem (FLT)  is a consequence of the  modularity theorem.
  • Elliptic functions  are doubly-periodic functions of a complex variable.
  • Elliptic curves:  Nonsingular cubics in the projective plane.
  • Modular forms.
  • Modularity theorem:  Every  elliptic curve  is  modular.

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Elliptic Curves  by  J.S. Milnes  (238 pages, 2006).
Lecture Notes on Elliptic Curves  by  Pete L. Clark   (Georgia, Fall 2012).

Elliptic curves and modular forms (41:01)  by  Michael L. Baker  (2015-06-11).


The Modularity Theorem
" Every Elliptic Curve is Modular "

 Pierre de Fermat  
 (1601-1665) (2017-07-06)   The Proof of Fermat's Last Theorem   (FLT)
Ken Ribet  showed that the  modularity conjecture  implied  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.  indeterminatereciprocity law  and also the first example of a non-abelian reciprocity law.

In 1955,  the Japanese mathematician  Yutaka Taniyama (1927-1958) made a wonderful conjecture which was refined in 1957 by his colleague  Goro Shimura (b.1930).  The conjecture was given its final form in 1967 by the Frenchman  André Weil (1906-1998).

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 FLT to the above (which was still a conjecture) by considering an hypothetical counterexample:

a p  +  b p   =   c p     for some odd prime  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...

That motivated a lonesome seven-year quest by  Andrew Wiles (b.1953)  to prove the modularity theorem, at least for  semistable elliptic curves  (which include Frey curves).  Surprisingly enough, Wiles was utimately successful  (on 1994-09-19)  thus producing a formal proof of  Fermat's Last Theorem !

The full  Modularity theorem  would finally be proved in 2001 using Wiles' ideas,  with contributions by  Christophe Olivier Breuil (b.1968, X1989) and three former doctoral students of  Andrew Wiles,  namely:  Richard Taylor (b.1962),  Fred Diamond (b.1964)  and  Brian Conrad (b.1970).

Fermat's Last Theorem:  The Theorem and Its Proof (1:36:40)  MSRI  (1993-07-28).
Fermat's Last Theorem (26:13)  by  Richard Pinch  (LMS, 1994).
The Bridges to Fermat's Last Theorem (27:52)   by  Ken Ribet   (Numberphile by Brady Haran, 2015-03-11).
Introducing the proof of Fermat's last theorem (16:06)   by  Randell Hyman   (2016-09-10).
Visualizing the Path from FLT to Calabi-Yau Spaces (1:04:52)   by  Andrew Hanson   (2011).

(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).

 Come back later, we're
 still working on this one...

Wikipedia :   Lattice

(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:  parabolashyperbolas  and  ellipses  (including circles).

Next up are the algebraic curves of degree 3,  which are called  cubics.  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 reduced to 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!

In modern terms...

 Come back later, we're
 still working on this one...

How How to Transform a Cubic (With a Rational Point) into Weierstrass Normal Form.
An elementary proof of the group law for elliptic curves   by  Stefan Friedl  (Rice, 2004-01-20).
A brief history of elliptic curves   by  Paul Hewitt  (2005-12-05).
An Introduction to the Theory of Elliptic Curves   by  Joseph H. Silverman  (Summer 2006).
Why is an elliptic curve a group? &nsp;   Harald Hanche-Olsen  (MathOverflow, 2009-11-26).
Wikipedia :   Elliptic curves   |   Cayley-Bacharach theorem (1886)
Abelian variety   |   Arithmetic of abelian varieties   |   Picard group

(2017-07-06)   Modular Forms

 Come back later, we're
 still working on this one...

Modular forms (1:20:52)  by  Andrew Snowden   (Math 679, Lecture 13, Michigan, 2013-10-22)
Wikipedia :   Modular forms   |   Cusp forms   |   Hecke algebra

(2017-07-06)   The Modularity Theorem
Known as the  Taniyama–Shimura–Weil conjecture, from  1967 to 2001.

 Come back later, we're
 still working on this one...

Wikipedia :   Modularity theorem

(2017-08-29)   Ramanujan's Tau Function
The Sato-Tate conjecture.

 Come back later, we're
 still working on this one...

Ramanujan tau function and the Sato-Tate conjecture (11:29).
Sato-Tate conjecture   |   Mikio Sato (1928-)   |   John Tate (1925-)

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