In Karl Rubin showed that for elliptic curves defined over an imaginary quadratic field with complex multiplication by , if the -series of the elliptic curve was not zero at , then the -part of the Tate-Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes. In Andrew Wiles, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor proved that all elliptic curves defined over the rational numbers are modular the Taniyama-Shimura theorem , which extends the previous two results to all elliptic curves over the rationals.

Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.

## Papers, Preprints and Lecture Notes

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## Elliptic Curve

AoPS Wiki. Page Article Discussion View source History. Background In Louis Mordell proved Mordell's Theorem : the group of rational points on an elliptic curve is finitely generated. An L-function can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime. History In the early s Peter Swinnerton-Dyer used the EDSAC computer at the University of Cambridge Computer Laboratory to calculate the number of points modulo denoted by for a large number of primes on elliptic curves whose rank was known. From these numerical results Bryan Birch and Swinnerton-Dyer conjectured that for a curve with rank obeys an asymptotic law Initially this was on the basis of somewhat tenuous trends in graphical plots; which induced a measure of scepticism in J.

### Literatura iz eliptičkih krivulja i njihove primjene u kriptografiji

Category : Number theory. Subscribe for news and updates. Login Cancel Stay logged in. Create a new account Lost your activation email? This is technically more sophisticated but I believe there are also implementations.

The best practical solution is to have someone else do the work. You can look up the curve in Cremona's tables , if it is not too large. It is misleading to say "algorithm", really. There are probabilistic algorithms, and then hard cases evidence of a point of infinite order that is hard to find.

I think of Andrew Bremner as one of the experts on the highly numerical side, and you should add his name to John Cremona's. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

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Algorithms for finding rational points on an elliptic curve? Ask Question.

### Your Answer

Asked 8 years, 11 months ago. Active 6 years, 7 months ago. Viewed 7k times. Andrej Bauer Andrej Bauer Section 4. But if one is only interested in the practical matter of finding points, rather than the well-foundedness of the methods, then there are algorithms. More precisely, if you believe the Birch and Swinnerton-Dyer conjecture, then there are efficient algorithms that are guaranteed to terminate.

Oct 13 '10 at I was talking about modular symbols, which were used by Cremona to generate his database of elliptic curves, but this has nothing to do with the present question. Sorry about that, I should stop posting so late at night! Jamie Weigandt Jamie Weigandt 2, 1 1 gold badge 23 23 silver badges 35 35 bronze badges. See the note here: warwick.

This makes life simpler for me, which his a good thing at least for me. I still use mwrank stand-alone myself hardly surprising. Secondly, Jamie's remark about mwrank in Sage being more up to date is simply nonsensical. I'm sorry was confused about that. Sorry about that, I should stop posting so late at night!

Jamie Weigandt Jamie Weigandt 2, 1 1 gold badge 23 23 silver badges 35 35 bronze badges. See the note here: warwick.

## [PDF] Rational Points on Modular Elliptic Curves - Semantic Scholar

This makes life simpler for me, which his a good thing at least for me. I still use mwrank stand-alone myself hardly surprising. Secondly, Jamie's remark about mwrank in Sage being more up to date is simply nonsensical. I'm sorry was confused about that. Robin Chapman Robin Chapman All the notation and equations in the presentation in Silverman-Tate really obscure what is going on, in my experience. Felipe Voloch Felipe Voloch The descents as in Robin's answer tell us that in order to find rational points on an elliptic curve, we better search on one of its torsors.

But in the end, we have to do some "brutal search" and that is where the crucial improvements in ratpoints are useful. But all of them only work when the analytic rank is 1. Most of this is implemented in magma and lots of it in sage. Inputing "import sage. Michael Stoll wrote a paper about this, but I can't find it online anymore it vanished from his website , so here is a temporary link: wstein.

If you type print EllipticCurve [a1,a2,a3,a4,a6]. For more about elliptic curves over the rationals in Sage, see this page: sagemath. Charles Matthews Charles Matthews They often rely on unproven conjectures, but they happen to work very well in practice. Also, probabilistic algorithms are referred to as "algorithms", just as deterministic ones, and I don't quite see what is misleading about this terminology.

Of course it is misleading here to mix up different kinds of algorithms. Brute force search will find a rational point on a curve if it is exists partial correctness. A correct algorithm is what people generally mean by something being "algorithmic". A good probabilistic algorithm is typically what is sought in computational number theory, and a good algorithm with correctness proof conditional on some known conjecture is of real interest.

What the heck are you talking about? Without making one of those assumptions, we still don't know.