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/
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