What Is The Definition Of The J-invariant Of An Elliptic Curve?

Why is there a group law on an elliptic curve?

• Following the immense success of I'm asking another "seed" question.  If this doesn't attract much attention I'll try to answer it myself, but I'd really rather hear from people who work with elliptic curves. Introductory undergraduate courses often demonstrate addition of real points on an elliptic curve by drawing lines.  This is unfortunate, in my opinion, as it gives no motivation as to how anyone would have come up with the idea in the first place*. Ideally, it would be nice to have a more comprehensive explanation which would be accessible to advanced undergraduates, and which might be presented in a survey course on number theory or cryptography.  It would be nice to include (at least heuristic) explanations for some or all of the following: How abelian varieties arise in calculus Why abelian varieties are tori Any part of the classification of algebraic groups** A nice picture of the real points of an elliptic curve sitting inside the complex plane modulo a lattice Anything about modular forms or the j-invariant, emphasizing the "modular" aspect. Assume whatever undergrad math courses are necessary, but not, say, deep familiarity with algebraic geometry or complex analysis. * informs me that the group laws on (real) elliptic curves were first developed in steps by Bachet, Fermat, and Newton.  Apparently explicit formulas were developed first, without the understanding that they corresponded to secants and tangents, and the geometric interpretation was realized later.  I feel this is pretty good evidence that this line of development is far from straightforward. ** E.g., the one-dimensional case, if it can be done on its own in an elementary manner.  Clearly the classification of linear algebraic groups by Lie algebras would be overkill, and only marginally relevant, here, but if you for instance had a nice easy way to see that an algebraic group is linear algebraic-by-abelian variety (sometimes known as Chevalley's theorem) then that would be nice.

• Answer:

  The question WHY there is a group law on the elliptic curves is one that can't really be answered in a very meaningful way. The reality is that there is a group law and we should focus on how to see this group law. Personally I think that if you are going to motivate the group law on an elliptic curve to undergraduates the right way to go is diophantine equations and rational points NOT real points.  In this case, the most natural place to start is with Diophantus’ work on quadratics. Diophantus noticed that if a quadratic equation in two variables has an rational solution, then it in fact has infinitely many rational solutions. Take for example the curve [math]C[/math] given by the equation [math]x^2 + 2xy + y^2 - x - 1 = 0[/math]. This equation has a rational solution at [math]P=(-1,1)[/math]. Now any line with a rational slope through the point [math]P[/math] will intersect [math]C[/math] in another rational point (unless it is tangent). Take for example the line [math]y=\frac{x}{2}+\frac{3}{2},[/math] this is the line of slope 1/2 through [math]P[/math]. Plugging this into the equation for [math]C[/math] you get [math]\frac{9x^2}{4}+\frac{7x}{2}+\frac{5}{x}=\frac{1}{4}(x+1)(9x+5)=0.[/math] So we have another rational point at [math]x=-\frac{5}{9}[/math]. A little bit of algebra shows that this point is [math]Q=\left(-\frac{5}{9},\frac{11}{9}\right)[/math]. I think it is a reasonable undergraduate assignment to show that given a curve [math]C[/math], of the form [math]a_0x^2+a_1y^2+a_2xy+a_3x+a_4y+a_5=0[/math] with a rational point [math]P=(\alpha,\beta)[/math], the line through [math]P[/math] with rational slope [math]m[/math] intersects [math]C[/math] in another rational point if the line is not tangent to [math]C[/math], and in fact that every rational point arises this way. Clearly the next natural question is, what about cubics? Diophantus went on to try the same trick on cubics, this time starting with two solutions and generating a third. It was only later that a slight modification of this method gave a group structure to the points on a cubic. Of course, Diophantus made no such claim, since the notion of a group didn't exist. Diophantus was more interested in how to generate rational solutions new rational solutions given old ones. As mathematics advance, people like become more interested in the structure of the rational points on a cubic rather than an algorithm to find all of them. This is when mathematicians first started applying the idea of a group to this situation. As far as using calculus to motivate elliptic curves, I would point to the following exercise (A.6.2) in Joseph Silverman's Diophantine Geometry: An Introduction "Consider the ellipse [math]\left(\frac{x}{a}\right)+\left(\frac{y}{b}\right) = 1[/math] with [math]a\geq b>0,[/math] and let [math]c^2=a^2-b^2[/math] with [math]c>0[/math]. Show that the computation of the arc length of this ellipse leads to the computation of an integral of the form [math]\displaystyle \int \frac{c^2u^2+b^2}{\sqrt{(1-u^2)(c^2u^2+b^2)}}\, du.[/math] Show the curve [math]w^2=(1-u^2)(c^2u^2+b^2)[/math] is an elliptic curve  except when [math]c=0[/math], that is, when the ellipse is a circle." I don't have a solution to this at the moment, but I suspect that the first part is just a matter of writing out the integral for arc length and picking the right [math]u[/math]-substitution. The second part, shouldn't be too hard either. I think one just has to be clever enough to find the appropriate transformation to put this in Weierstrass form. I would think a careful web search would yield an answer to this! If you are interested let me know in the comments and I will try and find something. I personally don't think that abelian varieties in general are a great subject for undergraduates. Showing that they are all complex tori requires some advanced machinery (I think). If you are hell bent on doing this, I would say the right perspective would be to consider the integral [math]\displaystyle \int\frac{1}{\sqrt{f(x)}}\, dx[/math] with the degree of [math]f(x)[/math] greater than or equal to 3. This integral is not path independent, one needs to make some branch cuts to make it well defined. Doing this and then glueing the two branches together gives the hyperelliptic curve [math]y^2=f(x)[/math] (or elliptic if the degree is 3 or 4) and one can then define its jacobian. This seems to me to be a lot for an undergrad, but maybe it depends on the group you are working with. A great source for this would be Silverman's book I mentioned above. Personally, I would teach the students about the group law using divisors and the class group of a curve. It is really the right way to think about the group law as associativety falls fairly easily once the groundwork has been laid. It also makes the group law sensible over finite fields which is very important if you are interested in cryptography. This idea of drawing tangent/secant lines loses its meaning in the case when you are working over finite fields. Talking about it in terms of the class group also opens the doors to a lot of interesting algebraic/arithmetic geometry including Riemann-Roch and the correct definition of genus. Of course I would limit myself to the case where my curve is smooth so that one only needs Weil divisors. Once you believe that elliptic curves are complex tori (genus 1 curves), which are can pretty easily be thought of as the complex plane modulo a lattice, the door is open to talking about modularity. One should start with lattices in the complex plane, then show that lattices modulo homothety (scaling) are essentially points in the upper half-plane. Once this is done, you can talk about how change of basis matrices have to be in [math]{\rm SL}(\mathbf{Z})[/math], define its action on the upper half-plane and mod out once again. The question then becomes how do you go from a point in the upper half-plane to an elliptic curve? Here is where you can start talking about the [math]j[/math]-map. I know I left out a lot of detail, but look how long this answer is! All of it is out there and if you are interested please let me know. I love elliptic curves and modular curves and I am always happy to talk about them. I will provide some links to good sources below, but if anything is unclear please feel free to comment. I didn't really get a chance to proof read this so I apologize for any typos etc. http://www.amazon.com/Elliptic-Modular-L-functions-Student-Mathematical/dp/0821852426/ref=sr_1_2?s=books&ie=UTF8&qid=1339788934&sr=1-2&keywords=elliptic+curve+alvaro http://livetoad.org/Courses/Documents/132d/Notes/history_of_elliptic_curves.pdf http://www.amazon.com/Rational-Points-Elliptic-Undergraduate-Mathematics/dp/0387978259 http://www.amazon.com/Arithmetic-Elliptic-Curves-Graduate-Mathematics/dp/0387094938/ref=pd_sim_b_1 http://www.amazon.com/Diophantine-Geometry-Introduction-Graduate-Mathematics/dp/0387989811/ref=sr_1_1?s=books&ie=UTF8&qid=1339788814&sr=1-1&keywords=diophantine+geometry+an+introduction http://calico.mth.muohio.edu/reza/sumsri/2002/notes.pdf