Other answers provide correct explanation to the associativity of Short answer: because it's a group lawcomplex torus. Yet,here's my Explanation below would take for a systematic explanation from the very start, trying to explain how it is possible to bring together several branches of mathematics to get a bit more complete picture of this beautiful topicas through many topics. The curve should be considered over complex numbers, then where it can be seen as a Riemann surface, so it's therefore a two-dimensional oriented closed variety. How to find out whether this particular one is a sphere, torus or something else? Just consider a two-fold covering onto $x$-axis and count the Euler characteristics as $-2 \cdot 2 + 4 = 0$ (don't forget the point at infinity.) So this is a torus; now a torus with complex structure can be always defined as a quotient $\mathbb C/\Lambda$. The , where $\Lambda$ is the lattice of periods $\Lambda$ . It can be constructed written as integrals $\int_\gamma \omega$ of any differential form $\omega$ over all elements $\gamma \in \pi_1$. The choice of differential form is unique up to $\lambda \in \mathbb C$. A complex map of a torus into itself that leaves fixed lattice $\Lambda$ fixed can be only given by a shift. Once you select a base point, these shifts are in one-to-one correspondence with points of $E$. We have unique distinguished point — infinity — so let's choose it as the base point. It follows that we now have an addition map* $(u, v) \to u\oplus v$, though defined purely algebraically so far. What would happen if you took a line through $u$ and $v$? By temporarily changing coordinates so that $u$ becomes the infinity pointand, one writes down that map as $(u, v) \mapsto -(u+v)$. The In the abovemethods seemed to require that , we work only worked over complex numbers, but in fact the final result is a proof of we proved associativity statement, expressed as which is a formal theorem about substitution of some rational expressions into others. Since it works over complex fields, it is required to work over all fields. (In any case, the big discovery of mid-20th century was that you actually can take much all of the intuition described above and apply it to the case of elliptic curves over arbitrary field; and for the topics discussed here, you can actually take all of the intuition.) Consider a line that passes through points $u$, $0$ and $-u$. This line is actually vertical, and $y$ is a well-defined function there which has two zeroes and one double pole at infinity. After a shift and multiplication of several such functions we'll be getting a meromorphic function on a complex torus with poles $p_i$ and zeroes $z_i$ having the property $\sum p_i = \sum z_i$. This method can give all such functions and only them; it's not hard to see that only meromorphic functions with this property are allowed hereon elliptic curve. For example, derivatives of Weierstrass $\wp$-functions\wp'$-functions are the ones that have triple pole at 0 and single zeroes at points $\frac12w_1, \frac12w_2, \frac12(w_1+ w_2)$ where $w_1, w_2$ are generators of $\Lambda$.
|
4 | reworded | ||
|
|
||||
|
|
3 | expanded | ||
|
The Other answers all provide correct explanation to the associativity of a group law. Yet,here's my take for a systematic explanation from the very start, trying to explain how different it is possible to bring together several branches of mathematics are brought together in to get a bit more complete picture of this beautiful topic. Jacobian of a curve (bonus 2)The formula above describes what types of functions are allowed on our curve. It is a good idea to organize this information into a curve: in this case, the information is that a single expression $p_1 + p_2 + \cdots + p_n - z_1 - \cdots - z_n$, considered a point of the curve, must vanish. For curves of higher genus, more relations are necessary; for $\mathbb C\mathbb P^1$, no relations beyond number of poles = number of zeroes are necessary. Those are relations in the group of classes of divisors (= Jacobian of a curve) mentioned in other answers. In particular, elliptic curves coincide with their Jacobian and that's another explanation for the additive law. |
||||
|
|
2 | expanded | ||
|
The answers all provide correct explanation to the associativity of a group law. Yet, the beauty of the topic is that it can be approached from different places. Here's here's my take for a systematic explanation from the very start, trying to explain how different branches of mathematics are brought together in this beautiful topic. Topological coversThe curve should be considered over complex numbers, then it can be seen as a Riemann surface, so it's a two-dimensional oriented closed variety. How to find out whether this particular one is a sphere, torus or something else? Just consider a two-fold covering onto $x$-axis and count the Euler characteristics as $-2 \cdot 2 + 4 = 0$ (don't forget the point at infinity.) Complex toriSo this is a torus; now a torus with complex structure can be always defined as a quotient $\mathbb C/\Lambda$. The lattice of periods $\Lambda$ can be constructed as integrals $\int_\gamma \omega$ of any differential form $\omega$ over all elements $\gamma \in \pi_1$. The choice of differential form is unique up to $\lambda \in \mathbb C$. Algebraic additionA complex map of a torus into itself that leaves fixed lattice $\Lambda$ can be only given by a shift. Once you select a base point, these shifts are in one-to-one correspondence with points of $E$. We have unique distinguished point — infinity — so let's choose it. It follows that we now have an addition map* $(u, v) \to u\oplus v$, though defined purely algebraically so far. Geometric meaningNow let's stop and ask ourselves: how to see this addition geometrically? For a start, consider map that sends $u$ to the third point of intersection with the line containing both $u$ and 0 (the infinity point). It's not hard to see that we fix 0 but change every class $\gamma$ in a fundamental group into $-\gamma$, so we must have the map $u\mapsto -u$ here. Group theory lawsWhat would happen if you took a line through $u$ and $v$? By temporarily changing coordinates so that $u$ becomes the infinity point and, one writes down that map as $(u, v) \mapsto -(u+v)$. Now if you took three points, there would be two different ways to add them; those would lead to $(u+v)+w$ and $u+(v+w)$ as complex numbers, which we know to be associative. QED. Logically provenThe above methods seemed to require that we work only over complex numbers, but in fact the final result is a proof of associativity statement, expressed as a formal theorem about substitution of some rational expressions into others. Since it works over complex fields, it is required to work over all fields. In any case, the big discovery of mid-20th century was that you actually can take much of the intuition described above to the case of arbitrary field; and for the topics discussed here, you can actually take all of the intuition. Analytic computations (bonus)Consider a line that passes through points $u$, $0$ and $-u$. This line is actually vertical, and $y$ is a well-defined function there which has two zeroes and one double pole at infinity. After a shift and multiplication of several such functions we'll be getting a meromorphic function on a complex torus with poles $p_i$ and zeroes $z_i$ having the property $\sum p_i = \sum z_i$. This method can give all such functions and only them; it's not hard to see that only meromorphic functions with this property are allowed here. For example, derivatives of Weierstrass $\wp$-functions are the ones that have triple pole at 0 and single zeroes at points $\frac12w_1, \frac12w_2, \frac12(w_1+ w_2)$ where $w_1, w_2$ are generators of $\Lambda$. |
||||
|
1 |
|
||
