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Math Help - The circle as a degenerate elliptic curve

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    MHF Contributor Bruno J.'s Avatar
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    The circle as a degenerate elliptic curve

    You may have heard of the fact that elliptic curves are groups. Perhaps you have little idea what an elliptic curve is, or in what sense we mean that it is a group. I'd like to discuss the group law for elliptic curves, as viewed from the degenerate case of the circle.

    First, recall that an (affine) elliptic curve C is a non-singuler curve of the form f(x,y)=0, where f \in \mathbb{R}[x,y] is a third-degree polynomial. Notice that any line \alpha x +\beta y = 0 which intersects the curve in two points intersects it in a third ; indeed, substituting the equation of the line into the equation defining C, we obtain a third-degree equation for which two roots are given, by assumption; therefore, there exists a third real root.

    We define a binary operation on C, as follows : for two points P,Q, we let P*Q be the third point of intersection of C with the line through P and Q. This operation trivially satisfies P*Q=Q*P and P*(P*Q)=Q. By choosing a point O on the curve, which we define as the identity element of the group, we define another operation by P+Q=O*(P*Q). Then it follows that P+Q=Q+P, and P+O=P. Thus C will have been made into a group if we can prove that + is associative. I'm not going to prove this; although it's not terribly difficult, it's certainly less obvious than proving that the other two axioms for a group are satisfied.

    Now let's take a look at the circle group. What does it mean to add points on the circle? It means to multiply them as complex numbers, or to add their angles with the positive x axis, or to compose the two plane rotations which they evidently describe. But there's a way to add points on the circle without any rotations. It's based on the theorem which states that two parallel lines which intersect the circle sandwich arcs of equal length. Using this theorem, we can "translate" an arc on the circle simply by drawing two parallel lines from its endpoints. So we pick a point, say 1=(1,0), to be the identity of the group. Now to add two points P and Q, we draw a line L from P to Q, and a second line L' through 1 and parallel to L; the intersection of L' with the circle is P+Q.

    Now suppose that we were adding points on the circle as if it were an elliptic curve. Suppose that there were a vertical line L, very distant to the right of the circle. To add P and Q, we would draw the line through P and Q, which would intersect the line somewhere very far. Then, from this point we would draw another line through 1, which would intersect the circle in P+Q. Now the two lines which we have drawn are almost parallel, provided that L is very distant. As L becomes more and more distant, the point P+Q thus approaches its traditional value.

    It would be interesting to back all of this up with some good ol' equations. Maybe I'll do that later if I'm bored. I just wanted to share.
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    One of the main applications of the group structure on an elliptic curve is in number theory. If two points P, Q on an elliptic curve have rational coordinates then so does P*Q. In particular, you can take Q=P, so that the line PQ is the tangent at P. Then you only have to know one rational point P on the curve in order to generate others. This is a powerful method for finding families of solutions to cubic diophantine equations.

    I wondered how that idea would play out on the circle. A rational point P = \bigl(\tfrac pr,\tfrac qr\bigr) on the unit circle is given by a primitive pythagorean triple (p,q,r). The tangent at P has gradient –p/q, and the line through (1,0) with that gradient meets the circle at the point (x,y) given by x=\tfrac{p^2-q^2}{p^2+q^2}, y=\tfrac{2pq}{p^2+q^2}, telling you that ( p^2-q^2, 2pq,p^2+q^2) is a primitive pythagorean triple.

    Nothing new there, of course, but it's a neat illustration of how the method works.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Opalg View Post
    One of the main applications of the group structure on an elliptic curve is in number theory. If two points P, Q on an elliptic curve have rational coordinates then so does P*Q. In particular, you can take Q=P, so that the line PQ is the tangent at P. Then you only have to know one rational point P on the curve in order to generate others. This is a powerful method for finding families of solutions to cubic diophantine equations.

    I wondered how that idea would play out on the circle. A rational point P = \bigl(\tfrac pr,\tfrac qr\bigr) on the unit circle is given by a primitive pythagorean triple (p,q,r). The tangent at P has gradient –p/q, and the line through (1,0) with that gradient meets the circle at the point (x,y) given by x=\tfrac{p^2-q^2}{p^2+q^2}, y=\tfrac{2pq}{p^2+q^2}, telling you that ( p^2-q^2, 2pq,p^2+q^2) is a primitive pythagorean triple.

    Nothing new there, of course, but it's a neat illustration of how the method works.
    Opalg,Bruno,

    I am familiar with the idea of curves and their uses in differential geometry, but never in number theory (although, of course we all know 'elliptic curves, FLT, etc.') could you further explain how they are used there? They also show up with, no small frequency, in the very little alg. geo I've studied.
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    MHF Contributor Bruno J.'s Avatar
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    That's a good question. I'm not sure I can give a complete answer but I can at least give a partial answer.

    What's a curve? A geometer would perhaps say that it's a continuous map from the real line to some other space, perhaps having certain specific properties such as smoothness. For number theory, this is obviously much too general of a definition. Historically, the first examples of curves were discovered as sets of points satisfying certain equations. It's not at all clear that this definition, in certain circumstances, will match the geometer's definition; it's rather a consequence of certain theorems of analysis such as the inverse function theorem or the like.

    Historically, much of mathematics has been preoccupied with finding solutions to certain equations. Number theory is concerned with those solutions which have a particularly nice form, such as rational, integral or algebraic solutions. That's when number theory and geometry first started dating : when we tried to find certain points on specific curves. Fermat's last theorem, although it was certainly not originally interpreted as a geometric statement, is really a statement about the set of rational points on a special curve (the Fermat curve f(x,y)=x^n+y^n-1=0.

    It turns out that the relationship between number theory and geometry is a very happy marriage. Elliptic curves were considered as algebraic objects as soon as they were discovered; they were born basically at the same time as the discovery of their group structure - that's why they became objects of interest so quickly.

    It turns out, however, that the geometric intuition is only a guide, and that the true nature of these mathematics is algebraic. There is no way to visualize an elliptic curve over a finite field - yet these curves provide an extremely important insight into their rational counterpart. André Weil formulated a series of extremely important conjectures (now theorems) regarding the structure of elliptic curves over finite fields - part of this theorem is actually a sort of Riemann hypothesis regarding the zeta functions associated to these curves, and it is now proven! And I don't need to go into the importance of the RH for number theory - it's certainly the most important unsolved problem in that field.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    That's a good question. I'm not sure I can give a complete answer but I can at least give a partial answer.

    What's a curve? A geometer would perhaps say that it's a continuous map from the real line to some other space, perhaps having certain specific properties such as smoothness. For number theory, this is obviously much too general of a definition. Historically, the first examples of curves were discovered as sets of points satisfying certain equations. It's not at all clear that this definition, in certain circumstances, will match the geometer's definition; it's rather a consequence of certain theorems of analysis such as the inverse function theorem or the like.

    Historically, much of mathematics has been preoccupied with finding solutions to certain equations. Number theory is concerned with those solutions which have a particularly nice form, such as rational, integral or algebraic solutions. That's when number theory and geometry first started dating : when we tried to find certain points on specific curves. Fermat's last theorem, although it was certainly not originally interpreted as a geometric statement, is really a statement about the set of rational points on a special curve (the Fermat curve f(x,y)=x^n+y^n-1=0.

    It turns out that the relationship between number theory and geometry is a very happy marriage. Elliptic curves were considered as algebraic objects as soon as they were discovered; they were born basically at the same time as the discovery of their group structure - that's why they became objects of interest so quickly.

    It turns out, however, that the geometric intuition is only a guide, and that the true nature of these mathematics is algebraic. There is no way to visualize an elliptic curve over a finite field - yet these curves provide an extremely important insight into their rational counterpart. André Weil formulated a series of extremely important conjectures (now theorems) regarding the structure of elliptic curves over finite fields - part of this theorem is actually a sort of Riemann hypothesis regarding the zeta functions associated to these curves, and it is now proven! And I don't need to go into the importance of the RH for number theory - it's certainly the most important unsolved problem in that field.
    Interesting. So what in particular do we study about these curves? For example, you mention Weil's conjectures which (as far as my understanding goes--i.e. a professor tried to explain it to me once) relates the number of solutions of an polynomial over a finite field (or all finite fields, I can't recall) the dimension of the cohomology rings of the associated varieties. So is that what is useful to us? Are computing the cohomology rings of these curves because they are really varieties?
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    Elliptic curves also have important applications in cryptography. That's not a subject I know anything about, but you can get some glimpse of it here.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Drexel28 View Post
    Interesting. So what in particular do we study about these curves? For example, you mention Weil's conjectures which (as far as my understanding goes--i.e. a professor tried to explain it to me once) relates the number of solutions of an polynomial over a finite field (or all finite fields, I can't recall) the dimension of the cohomology rings of the associated varieties. So is that what is useful to us? Are computing the cohomology rings of these curves because they are really varieties?
    Haha, I don't know enough to even pretend that I could begin answering any question about Weil's conjectures. It's certainly something I'd like to learn about some day, though.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Haha, I don't know enough to even pretend that I could begin answering any question about Weil's conjectures. It's certainly something I'd like to learn about some day, though.
    Haha, I know what you mean. When the dude tried to explain it to me he's like "So remember how \displaystyle \bigoplus_{k\in\mathbb{Z}}H^k_\bullet(X;\mathbb{C}  )\cong\int^{\bigoplus}}fd\omega" and I was like "Of course. "

    Well, how did you become interested in elliptic curves? What are you studying them right now? Like what subject?
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    MHF Contributor Bruno J.'s Avatar
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    I became interested in elliptic curves by studying the theory of elliptic functions and elliptic integrals. I had the opportunity to study elliptic curves over \mathbb{C}, as Riemann surfaces. I'm now studying elliptic curves over other types of fields more closely related to number theory. Over \mathbb{C}, the group structure of an EC is not very rich - it's always the torus group. The analytic/geometric structure is what's interesting over \mathbb{C}: we're mostly interested in determining when two curves are isomorphic as Riemann surfaces, and finding parametric families of curves which may help identify certain symmetries of the curves - this leads to the study of modular forms.

    Over smaller fields, the group structure of EC's is much more complicated and much more interesting. That's what I'll be studying for the next few months!
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    I became interested in elliptic curves by studying the theory of elliptic functions and elliptic integrals. I had the opportunity to study elliptic curves over \mathbb{C}, as Riemann surfaces. I'm now studying elliptic curves over other types of fields more closely related to number theory. Over \mathbb{C}, the group structure of an EC is not very rich - it's always the torus group. The analytic/geometric structure is what's interesting over \mathbb{C}: we're mostly interested in determining when two curves are isomorphic as Riemann surfaces, and finding parametric families of curves which may help identify certain symmetries of the curves - this leads to the study of modular forms.

    Over smaller fields, the group structure of EC's is much more complicated and much more interesting. That's what I'll be studying for the next few months!
    Interesting. Is this study part of a formal thing or on your own?


    Also, would you say that elliptic curves would be first encountered in a complex analysis course opposed to a say algebraic number theory or algebraic geometry course?
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    MHF Contributor Bruno J.'s Avatar
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    I first encountered elliptic curves in a graduate complex analysis course. I had the chance to do a research project last year on special types of moduli spaces for elliptic curves, so I'd say it's both part of a formal thing and a personal endeavour. I'm starting to have a pretty good understanding of certain things now and I really love it!

    You could approach the topic from either complex analysis, algebraic geometry, algebraic number theory or analytic number theory. The topic is quite far-reaching!
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    how can it be used to prime factor integers?
    Factorization using the Elliptic Curve Method
    Last edited by Bruno J.; June 15th 2011 at 10:07 AM.
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    MHF Contributor Bruno J.'s Avatar
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    Re: The circle as a degenerate elliptic curve

    There's a good explanation of the method in Silverman & Tate's "Rational points on elliptic curves".
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: The circle as a degenerate elliptic curve

    Quote Originally Posted by Bruno J. View Post
    You may have heard of the fact that elliptic curves are groups. Perhaps you have little idea what an elliptic curve is, or in what sense we mean that it is a group. I'd like to discuss the group law for elliptic curves, as viewed from the degenerate case of the circle.

    First, recall that an (affine) elliptic curve C is a non-singuler curve of the form f(x,y)=0, where f \in \mathbb{R}[x,y] is a third-degree polynomial. Notice that any line \alpha x +\beta y = 0 which intersects the curve in two points intersects it in a third ; indeed, substituting the equation of the line into the equation defining C, we obtain a third-degree equation for which two roots are given, by assumption; therefore, there exists a third real root.

    We define a binary operation on C, as follows : for two points P,Q, we let P*Q be the third point of intersection of C with the line through P and Q. This operation trivially satisfies P*Q=Q*P and P*(P*Q)=Q. By choosing a point O on the curve, which we define as the identity element of the group, we define another operation by P+Q=O*(P*Q). Then it follows that P+Q=Q+P, and P+O=P. Thus C will have been made into a group if we can prove that + is associative. I'm not going to prove this; although it's not terribly difficult, it's certainly less obvious than proving that the other two axioms for a group are satisfied.

    Now let's take a look at the circle group. What does it mean to add points on the circle? It means to multiply them as complex numbers, or to add their angles with the positive x axis, or to compose the two plane rotations which they evidently describe. But there's a way to add points on the circle without any rotations. It's based on the theorem which states that two parallel lines which intersect the circle sandwich arcs of equal length. Using this theorem, we can "translate" an arc on the circle simply by drawing two parallel lines from its endpoints. So we pick a point, say 1=(1,0), to be the identity of the group. Now to add two points P and Q, we draw a line L from P to Q, and a second line L' through 1 and parallel to L; the intersection of L' with the circle is P+Q.

    Now suppose that we were adding points on the circle as if it were an elliptic curve. Suppose that there were a vertical line L, very distant to the right of the circle. To add P and Q, we would draw the line through P and Q, which would intersect the line somewhere very far. Then, from this point we would draw another line through 1, which would intersect the circle in P+Q. Now the two lines which we have drawn are almost parallel, provided that L is very distant. As L becomes more and more distant, the point P+Q thus approaches its traditional value.

    It would be interesting to back all of this up with some good ol' equations. Maybe I'll do that later if I'm bored. I just wanted to share.


    It reminded me this lecture: go to 15:20.
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    Re: The circle as a degenerate elliptic curve

    thank you for posting that video! it was very entertaining.
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