Talk:Elliptic curve

message: A quick note : I find the animating demonstration of associativity to be fascinating.. but it seems to me the labels of the points are incorrect?  The 'zero' is fixed (good), but then, for example 'a' + 'b' should be further from the origin than 'a' or 'b' alone?  Can some kind wiki-helper get in touch with the .GIF author and double check everything is in order?  — Preceding unsigned comment added by 210.55.212.205 (talk) 20:54, 16 December 2014 (UTC)
thread_title: label mistake in associativity, ellipticgroup.gif

message: I'm confused here. In the text, it states:
:By adding a point "at infinity", we obtain the projective version of this curve; every straight line intersects this curve in three points (if the line is tangent to the curve at a point, then that point is counted twice). It is then possible to introduce a group operation on the curve with the following property: if a straight line intersects the curve at the points P, Q and R, then P + Q + R = 0 in the group.
But, for example, the x-axis intersects the curve y2 = x3 - x + 1 at only 1 point (call it P); if we consider the other two points as being infinity, this seems to require that P + inf + inf = 0. However, the x-axis is not alone in this property; several lines parallel to the x-axis also intersect at only 1 point, if we select one of these and call the point of intersection Q, then Q + inf + inf = P + inf + inf = 0 implies P = Q. Bzzzt!!!! Is the group operation restricted to those lines which actually intersect at at least two points (where tangency counts as 2 points)? Chas zzz brown 02:00 Jan 22, 2003 (UTC)
My statement above was wrong: it only that way only for algebraically closed base fields. You're right: if the fields isn't algebraically closed, we only consider lines that are tangent to the curve or intersect it in two points. AxelBoldt 17:23 Jan 22, 2003 (UTC)
----
I'd like to see a bit more about the group aspect of elliptic curves over finite fields (with an eye towards ECC); especially more describing an algebraic (as opposed to geometric) approach to calculating P + Q (see  for a nice explanation). Should that be done at this article, at the article for elliptic curve cryptography, or at a new article? Chas zzz brown 20:04 Feb 7, 2003 (UTC)
I think it can still fit here. Can't we give the general group law formula which works for all base fields? AxelBoldt 07:37 Feb 9, 2003 (UTC)
:Perhaps the best way to handle this would be to emphasise that the geometric picture is only valid for the real field, and then to explain that nevertheless we can make sense of things over any field by working out some algebraic formulation. The really important point here is that if two given points have rational coordinates (i.e. belong to some field), then so does the third point. This is why the group operation makes sense when defined over a given field. --Dmharvey 21:00, 28 May 2005 (UTC)
thread_title: 2003 material

message: —Sean κ. + 23:33, 27 May 2005 (UTC)
thread_title:  nother picture 

message: The article describes elliptic curves as being of the form y^2 = x^3 + ax + b.  However, in Dan Bernstein's Elliptic Curve Diffie Hellman library, he uses the curve y^2 = x^3 + Ax^2 + x where A = 486662.  Is this still an elliptic curve?
:If you think about the substitution x = x′ + c you see that there is some freedom here. That is, provided you can divide by 3 in the field, you can set c so that the Bernstein form has the coefficient of x′2 changed to 0. Charles Matthews 15:58, 26 October 2005 (UTC)
thread_title:  Form of the polynomial 

message: Does anyone know why these curves are called elliptic curves? My apologies if I missed it in the article. -Monguin61 01:47, 10 December 2005 (UTC)
replies: text: The connection is: elliptic curve => Weierstrass elliptic function (parameterises an elliptic curve) => elliptic integral => arc length of an ellipse. I tend to think of the name "elliptic curve" as a historical accident, and a poor name for the object it describes. Dmharvey 01:54, 10 December 2005 (UTC)
thread_title:  History of the term 

message: What do we call those?
I've heard them being called Elliptic curves.
Or non-singular curves on this form
y^2 z + axyz + byz^2 = x^3 + cx^2 z + d x z^2 + e z^3
being called elliptic curves...
Your definition appears to be different.
: Actually, those ARE elliptic curves. But every elliptic curve on a field with Characteristic different than two or three is isomorphic (in some Morphisms that preserve the Group Structure of the curve) to a curve like the ones defined in the article. So by giving the other definition, we didn't miss any information. BrunoX 02:03, 7 December 2006 (UTC)
thread_title:  Non-singular cubic curves? 

message: Can someone with knowledge disambig varieties in the Isogeny section? Thanks. Simon12 02:26, 1 July 2006 (UTC)
replies: text: I took my best guess. Change if incorrect. Simon12 03:15, 1 July 2006 (UTC)
thread_title:  DAB help needed 

message: I am an arithmetic geometer working in the field(s?) of elliptic curves and curves of genus one.  Here are my 
reactions to the article as currently written (it is quite good overall):
The basic definition -- "an elliptic curve is..." is close but needs even more care: first of all, 
if one is going to define it in earnest using the language of algebraic / arithmetic geometry (BTW, it is 
not clear to me that one should do this right away; more people would understand the y^2 = x^3+ax+b definition, 
which is not wrong, just overly concrete for some purposes), you are missing the proviso "geometrically 
connected."  More seriously, you are missing a mention of the field of definition of the elliptic curve.  As 
stated above, for about 100 years it has been the case that elliptic curves are interesting only over 
non-algebraically closed fields, so the dependence on the field should be made clear from the start.  Again, a 
reasonable choice would be to say a bit first about elliptic curves over the rational field and then start 
again with a definition over an arbitrary field (with the longer Weierstrass equation).  
Nitpicking point: the definition of an elliptic curve in short Weierstrass form is not insufficiently "precise"; 
it is simply not the definition of an elliptic curve in char. 2 or 3 that agrees with the above geometric 
definition (i.e., the problem is not precision; it's simply a matter of a statement which is no longer true).  
The paragraph beginning "If y^2 = P(x)..." is a bit awkwardly written.  In particular, somewhere in this 
paragraph the fact that an elliptic curve must have a rational point is forgotten.  The point of the Weierstrass 
form is that it comes equipped with a canonical rational point (the inflection point at infinity).  The other 
forms -- hyperelliptic quartic, plane cubic and intersection of two space quadrics -- are important alternate 
ways of presenting smooth curves of genus one, but in these other forms a rational point is not guaranteed 
(rather, a rational divisor of degree 2, 3, or 4, respectively).    
Saying that the proof of FLT is by Wiles assisted by Taylor is not ideal -- the work of Taylor and Wiles was 
indeed spectacular, but it also built on important work of many other mathematicians: e.g. Ribet, who proved 
that Taniyama-Shimura implies FLT.  The attribution of FLT is a complicated story which can be left to the 
article itself; no need to try to wing it here.
"By adding a 'point at infinity', we obtain the projective version of this curve" This could be better phrased.  
The projectivization of an affine curve is an easy general construction: we just homogenize the defining 
polynomial by inserting in each monomial term whatever power of Z brings the total degree of each term up 
to the maximum of the total degrees of the original terms (in this case 3, of course).  Surely this is 
described somewhere onsite?  (If not, it should be!)  Then the fact that the line at infinity intersects a 
Weierstrass cubic in a single point can be verified.  My point is that the way it is said it sounds like 
one knew in advance that there was one point to add.  This is really not the case, since e.g. one could 
work with a non-Weierstrass cubic endowed with a rational point, and then the projectivization could add 
as many as three points.  
In your definition of the group law: you never actually say what P+Q _is_, you just give defining properties.  
Why not say that if R is the third intersection point of P and Q with the curve, then P+Q is the third 
intersection point of O and R with the curve (and illustrate this with a picture)?
"One can check that this turns the curve into an abelian group, and thus into an abelian variety."  Bad -- an 
abelian variety is a commutative group variety but not conversely (think of the additive group or the 
multiplicative group, e.g.).  The sentence could read, "One can check that the K-rational points on E 
form an abelian group under +.  [insert some indication that the associativity is the sticky point!]  
Moreover, as a map from E x E to E, the addition law is "algebraic", i.e., given by rational functions on E, 
so endows E with the structure of an algebraic group (and indeed, since E is projective and geometrically 
connected, an abelian variety."  Or you could not say all these things -- it's not really necessary.
"The above group law can be described algebraically as well as geometrically."  I object to this dichotomy 
here and elsewhere in the article: the above description of the group law is both algebraic _and_ geometric 
(and also arithmetic, since it takes fields of rationality into account).  It is not necessary to perform 
the construction first over the real numbers and then describe it separately -- it makes sense over any field 
using simple, purely algebraic definitions of partial derivative, tangent line, etc.  "Algebraically" and 
"using the following explicit equations" are not synonymous!
Elliptic curves over the complex numbers: "curious property of Weierstrass's elliptic functions..." "Curious" 
is unencyclopedic.  Anyway, it is not explained.  "Looks like" |-> "is homeomorphic to" (with a link).  The 
reader who has made it this far will not be confused.  ... "then the corresponding elliptic curves are 
isomorphic"  And conversely!  
"The isomorphism classes can be understood in a simpler way..." "Simpler" is POV.
"The complex numbers are the splitting field for polynomials" -> "the complex numbers are algebraically closed"
Drop the name "Legendre form" for the y^2 = x(x-1)(x-lambda) and give a reference (say, to Silverman) for 
all these formulas.
"The uniformization theorem states" |-> "The uniformization theorem implies."  (It says other things as well.)
Elliptic curves over a general field: again, I wish this would be more gracefully incorporated into the 
above.  (One way to do it is to do it over Q in a way that works over an arbitrary field, and then later 
to announce this.) 
"One typically takes the curve to be..." This is not quite right; a curve is a curve, in the sense of 
arithmetic geometry.  It is not simply the set of its points over the algebraic closure (although the 
set of points together with all morphisms from it are sufficient to recover the algebraic structure).  
It is better not to say anything about this subtle point than to get it a little bit wrong.
Isogeny of elliptic curve: the term "basepoint" (should be "base point") has not been used above.  (Better, 
e.g. "the origin", or "the neutral element".)  About the homomorphism property of an isogeny: moreover, 
the homomorphism is surjective over the algebraic closure.
"no general algorithm is known..." Elaborate on this point.  There is an algorithm which we use in practice, 
which will work provided Shafarevich-Tate groups are finite.  (The point is that having an "algorithm" that 
has not been proved to terminate in all cases is not the same as not having any idea at all how to start 
computing.)  
"A formula for this rank is given by..." This is true, but the other terms in the formula are difficult both 
theoretically and computationally as well.  
"This fact can be understood and proven with the help of some general theory..." Well, okay.  I wonder whether 
the people who are conversant (or even potentially conversant) with etale cohomology are really looking on 
Wikipedia to find out how to prove Hasse's bound.  Of course Hasse's proof was simpler, and there is also a 
relatively simple proof in Silverman's book: a reference here would be nice.
"The number of points on a specific curve..."  It would be hard to compute the number of points on a nonspecific 
curve, wouldn't it? :)
It would be nice also to have a section on elliptic curves over local fields and reduction theory, as well as a 
little bit about complex multiplication.
Well, I can help out with these changes, but not tonight. 128.192.134.50 06:58, 14 August 2007 (UTC)
replies: text: Why does an elliptic curve have to have a rational point?  If it doesn't, it's not called an elliptic curve?? Tkuvho (talk) 13:30, 2 December 2010 (UTC)
thread_title:  Some comments 

message: Did we lose something along the way? The O in the first sentence doesn't seem to help with the definition.  Perhaps there was some characteristic of O that is missing? (John User:Jwy talk) 23:00, 19 October 2008 (UTC)
thread_title: First sentence

message: I'm not an algebraic geometer/number theorist, but this looks wrong: "For prime numbers ℓ not dividing N, the coefficient \scriptstyle a(\ell) of the form equals ℓ – the number of solutions of the minimal equation of the curve modulo ℓ." Shouldn't it be "...equals aℓ – the number of solutions..."? If so, probably needs a reference  — Preceding unsigned comment added by 203.24.207.178 (talk) 06:10, 31 July 2012 (UTC)
thread_title: A typo?

message: why didn't you like my edit?
The link to Tunnell, as it stands, isn't useful. The referenced Tunnell isn't even one of those mentioned in the disambiguation page, whereas I proposed an useful link.
At least an explanation is in order instead of just bluntly undoing.  — Preceding unsigned comment added by 89.247.124.39 (talk) 13:21, 24 December 2012 (UTC)
replies: text: I don't understand this.   made this change to link to Tunnell's theorem and that link is still there.  I assume that  is the same person?  My subsequent change was quite unrelated.  It would help if the anonymous editor(s) would (1) explain exactly what it is they are objecting to here, (2) address personal messages to other editors on User talk pages not article talk pages, and (3) consider acquiring their own editor name to facilitate communication with other users.  Deltahedron (talk) 14:00, 24 December 2012 (UTC)
replies: text: Not a problem.  But please do consider Wikipedia:Why create an account?.  Deltahedron (talk) 17:28, 26 December 2012 (UTC)
thread_title:  Hey, Deltahedron... 

message: Section 3 seems to contain some fundamental mistakes. The formulas linking $g_2$, $g_3$ and $\Delta$ to $\lambda$ are clearly wrong, since the former are modular forms (transforming with a non-trivial weight) and the latter a modular function (transforming without a weight). I am not sure how to fix this.129.16.128.185 (talk) 14:07, 9 June 2016 (UTC)
thread_title:  Mistakes 

message: Under the section Elliptic curves over the complex numbers, this passage appears:
"the Weierstrass functions are naturally defined on a torus  T = C/Λ. This torus may be embedded in the complex projective plane by means of the map
:z \mapsto [1 :\wp(z) : \wp'(z)]
This map is a group isomorphism, carrying the natural group structure of the torus into the projective plane."
But it is not clear what it means to say that the map z \mapsto [1 :\wp(z) : \wp'(z)] is a group isomorphism.  
The projective plane does not carry a group structure of its own. So in what sense does the image of this map z \mapsto [1 :\wp(z) : \wp'(z)] even have a group structure?  
Of course, any bijection from a group to a set can be used to transfer the group structure of the group to one on its image. But in this case it is confusing to call that an isomorphism (even though it is one): It is just a map used to copy a group from one set onto another set.
I hope someone more knowledgeable than I will improve the exposition here.Daqu (talk) 19:07, 5 September 2016 (UTC)
replies: text: These sentences are talking about the relation between the torus C / Λ and the cubic curve y^2 = 4x^3 - g_2x - g_3.  I edited the article; hopefully it is clearer.  Ozob (talk) 02:53, 6 September 2016 (UTC)
replies: text: Thank you, Ozob!  It looks much clearer now.Daqu (talk) 07:41, 6 September 2016 (UTC)
thread_title:  Statement needs elaboration (or deletion) 

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thread_title:  External links modified 

message: There is the following mistake in the gif:
The "minus points", e.g. -(a+b), are incorrect. The labeling is based on the misconception that "three points on a line add up to 0". This is however only true if 0 is itself an inflection point.
That it is not true otherwise can easily be seen by considering the tangent at 0 itself.
The fix is that first the "minus labels" should be deleted and then also a line from 0 trough the middle point (currently labeled -(a+b+c)) should be drawn. The third intersection is then a+b+c.  — Preceding unsigned comment added by Claus from Leipzig (talk • contribs) 20:22, 21 June 2017 (UTC)
replies: text: I had missed your comment when I restored the animation a few days ago, sorry. I think it would be best to remove the section Elliptic curve#Associativity completely until the animation gets fixed as it really doesn't make much sense without the animation. See section #Associativity above, too. Tea2min (talk) 07:05, 2 August 2017 (UTC)
thread_title:  On the "minus points" in the gif 

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thread_title:  External links modified 

message: Surely the formulas for g_2 and g_3 here need to be multiplied by something. The modular lambda function :H -> C is invariant under all of \Gamma(2) but g_2 and g_3 are not, so they cannot be polynomials in \lambda. 
I think for instance the formula for g_3 needs to be multipled by the modular form -4 \pi^6 \theta_3(0,1+\tau)^{12} (after replacing \lambda by 1-\lambda).
It's nicely written and I don't advocate increasing the complexity, but surely simplifying things to the point of incorrectness can't have been the best choice. Createangelos (talk) 09:56, 11 February 2019 (UTC)
thread_title:  Elliptic curves over complex numbers 

message: Why is the coefficient of the x3 term shown as 4? Doesn't 4=1 in a field of characteristic 3?
MeanStandev (talk) 18:07, 27 June 2020 (UTC)
thread_title:  Question about elliptic curve over field of characteristic 3 

message: In Elliptic curves over the rational numbers:The structure of rational points the largest known rank is ">=28". There is no mention of how it is then determined how this might become "=28". Is this worth mentioning?
Darcourse (talk) 04:50, 26 April 2022 (UTC)
thread_title:  How is a rank determined to be exact? 

message: The reference for the book "Algorithms for Modular Elliptic Curves" links the author to the article on the Maltese politician John Joseph Cremona (https://en.wikipedia.org/wiki/John_Cremona).  The author is in fact John E. Cremona, emeritus professor at the University of Warwick (https://warwick.ac.uk/fac/sci/maths/people/staff/john_cremona/, https://johncremona.github.io/).  — Preceding unsigned comment added by Autoplunger (talk • contribs) 03:59, 13 May 2022 (UTC)
replies: text: Thanks, I fixed it. Prame tan (talk) 07:48, 3 April 2023 (UTC)
thread_title:  Wrong John Cremona in references 

message: This looks nice, but detailed discussion of the proof and its implications seems like it should belong on the Mordell–Weil theorem page rather than in a subsection here. To be specific, I think the part from "The proof of the theorem involves two parts." to "The theorem however doesn't provide a method to determine any representatives of E(Q)/mE(Q)." can be moved. Kclisp (talk) 17:12, 26 March 2023 (UTC)
thread_title:  The structure of rational points: Move proof of Mordell–Weil theorem