Elliptic Curve Resources

Date: 05/04/2007 at 12:20:03
From: Jaakko
Subject: Diophantine equations

I have heard there is no general algorithm to determine the rank of a
general elliptic curve.  If one proves the Swinnerton-Dyer conjecture,
will it give an algorithm to determine the rank of a given elliptic curve?

I read about Baker's theorem on page 261 in Silverman's book the
arithmetic of elliptic curve.  How do we define E(Q)?  Will that 
theorem give an upper bound for searching rational points of
polynomial form Diophantine equations?

I have heard that understanding elliptic curves requires some
knowledge about algebraic geometry, algebraic number theory, and class
field theory.  I'm interested in those fields so what would be a good
book to start studying those?

Date: 05/12/2007 at 13:13:00
From: Doctor Vogler
Subject: Re: Diophantine equations

Hi Jaakko,

Thanks for writing to Dr. Math.  There are algorithms for determining
the rank of an elliptic curve.  There are even programs available that
do just that.  One is mwrank (as in Mordell-Weil Rank) by John
Cremona, which you can download at

    http://www.maths.nott.ac.uk/personal/jec/ 

This uses the method of descent to determine the rank, but while it
works just fine for most curves, it is known to fail for some curves,
which probably disqualifies it from being called a "general
algorithm."  That is, it works better in practice than it does in
theory.  There are other algorithms that work better in theory but
which have very long runtimes in the sense that you have to perform
large numbers of computations, sometimes more than is feasible.

Silverman is a standard reference book on elliptic curves, but it is
written at a pretty advanced level, assuming a good knowledge of the
much more complicated field of algebraic geometry.  If you want to
learn about elliptic curves without trying to learn a lot of
unnecessary algebraic geometry, then I would recommend the book
"Elliptic Curves" by Knapp.  By the way, E(Q) is the group of rational
points on the elliptic curve E, under the elliptic curve's group law.

Unfortunately, learning the algebraic geometry is a lot harder. 
Usually, people recommend "Algebraic Geometry" by Hartshorne, but that
book is *very* hard to follow.  Another student suggested Masayoshi
Miyanishi's "Algebraic Geometry" (Translations of Mathematical
Monographs 136) as a more readable starting point.  I think I would
recommend starting with a book on classical algebraic geometry from
Robert J Walker's "Algebraic Curves".  This should help you to
understand what the modern algebraic geometry is aiming for.  Then you
can transition to modern algebraic geometry with something like David
M. Goldschmidt's "Algebraic Functions and Projective Curves" or Joe
Harris's "Algebraic Geometry: A First Course."

If you want to learn about algebraic number theory with a slant toward
algebraic geometry, then a good book would probably be Dino
Lorenzini's "An Invitation to Arithmetic Geometry."  It develops the
two topics in parallel.

If you have any questions about this, please write back.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/15/2007 at 10:50:29
From: Jaakko
Subject: Thank you (Diophantine equations)

Thanks for an excellent answer.  I'm interested to see those
algorithms to determine the rank of a given elliptic curve.  Where can
I find some information for those?

Date: 05/18/2007 at 21:10:09
From: Doctor Vogler
Subject: Re: Diophantine equations

Hi Jaako,

I mentioned "Elliptic Curves" by Knapp.  That has some of it in there.
You might also be interested in the descriptions by John Cremona, the
writer of mwrank.  You can find some papers on his web page at

    http://www.maths.nott.ac.uk/personal/jec/ 

I would recommend starting at one of those two places, but there are
plenty of other books and papers you could read too.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/