You are currently browsing the tag archive for the ‘elliptic curves’ tag.
Previous set of notes: Notes 2. Next set of notes: Notes 4.
On the real line, the quintessential examples of a periodic function are the (normalised) sine and cosine functions , , which are -periodic in the sense that
By taking various polynomial combinations of and we obtain more general trigonometric polynomials that are -periodic; and the theory of Fourier series tells us that all other -periodic functions (with reasonable integrability conditions) can be approximated in various senses by such polynomial combinations. Using Euler’s identity, one can use and in place of and as the basic generating functions here, provided of course one is willing to use complex coefficients instead of real ones. Of course, by rescaling one can also make similar statements for other periods than . -periodic functions can also be identified (by abuse of notation) with functions on the quotient space (known as the additive -torus or additive unit circle), or with functions on the fundamental domain (up to boundary) of that quotient space with the periodic boundary condition . The map also identifies the additive unit circle with the geometric unit circle , thanks in large part to the fundamental trigonometric identity ; this can also be identified with the multiplicative unit circle . (Usually by abuse of notation we refer to all of these three sets simultaneously as the “unit circle”.) Trigonometric polynomials on the additive unit circle then correspond to ordinary polynomials of the real coefficients of the geometric unit circle, or Laurent polynomials of the complex variable .What about periodic functions on the complex plane? We can start with singly periodic functions which obey a periodicity relationship for all in the domain and some period ; such functions can also be viewed as functions on the “additive cylinder” (or equivalently ). We can rescale as before. For holomorphic functions, we have the following characterisations:
Proposition 1 (Description of singly periodic holomorphic functions)In both cases, the coefficients can be recovered from by the Fourier inversion formula for any in (in case (i)) or (in case (ii)).
- (i) Every -periodic entire function has an absolutely convergent expansion where is the nome , and the are complex coefficients such that Conversely, every doubly infinite sequence of coefficients obeying (2) gives rise to a -periodic entire function via the formula (1).
- (ii) Every bounded -periodic holomorphic function on the upper half-plane has an expansion where the are complex coefficients such that Conversely, every infinite sequence obeying (4) gives rise to a -periodic holomorphic function which is bounded away from the real axis (i.e., bounded on for every ).
Proof: If is -periodic, then it can be expressed as for some function on the “multiplicative cylinder” , since the fibres of the map are cosets of the integers , on which is constant by hypothesis. As the map is a covering map from to , we see that will be holomorphic if and only if is. Thus must have a Laurent series expansion with coefficients obeying (2), which gives (1), and the inversion formula (5) follows from the usual contour integration formula for Laurent series coefficients. The converse direction to (i) also follows by reversing the above arguments.
For part (ii), we observe that the map is also a covering map from to the punctured disk , so we can argue as before except that now is a bounded holomorphic function on the punctured disk. By the Riemann singularity removal theorem (Exercise 35 of 246A Notes 3) extends to be holomorphic on all of , and thus has a Taylor expansion for some coefficients obeying (4). The argument now proceeds as with part (i).
The additive cylinder and the multiplicative cylinder can both be identified (on the level of smooth manifolds, at least) with the geometric cylinder , but we will not use this identification here.
Now let us turn attention to doubly periodic functions of a complex variable , that is to say functions that obey two periodicity relations
for all and some periods , which to avoid degeneracies we will assume to be linearly independent over the reals (thus are non-zero and the ratio is not real). One can rescale by a common scaling factor to normalise either or , but one of course cannot simultaneously normalise both parameters in this fashion. As in the singly periodic case, such functions can also be identified with functions on the additive -torus , where is the lattice , or with functions on the solid parallelogram bounded by the contour (a fundamental domain up to boundary for that torus), obeying the boundary periodicity conditions for in the edge , and for in the edge .Within the world of holomorphic functions, the collection of doubly periodic functions is boring:
Proposition 2 Let be an entire doubly periodic function (with periods linearly independent over ). Then is constant.
In the language of Riemann surfaces, this proposition asserts that the torus is a non-hyperbolic Riemann surface; it cannot be holomorphically mapped non-trivially into a bounded subset of the complex plane.
Proof: The fundamental domain (up to boundary) enclosed by is compact, hence is bounded on this domain, hence bounded on all of by double periodicity. The claim now follows from Liouville’s theorem. (One could alternatively have argued here using the compactness of the torus .
To obtain more interesting examples of doubly periodic functions, one must therefore turn to the world of meromorphic functions – or equivalently, holomorphic functions into the Riemann sphere . As it turns out, a particularly fundamental example of such a function is the Weierstrass elliptic function
which plays a role in doubly periodic functions analogous to the role of for -periodic real functions. This function will have a double pole at the origin , and more generally at all other points on the lattice , but no other poles. The derivative of the Weierstrass function is another doubly periodic meromorphic function, now with a triple pole at every point of , and plays a role analogous to . Remarkably, all the other doubly periodic meromorphic functions with these periods will turn out to be rational combinations of and ; furthermore, in analogy with the identity , one has an identity of the form for all (avoiding poles) and some complex numbers that depend on the lattice . Indeed, much as the map creates a diffeomorphism between the additive unit circle to the geometric unit circle , the map turns out to be a complex diffeomorphism between the torus and the elliptic curve with the convention that maps the origin of the torus to the point at infinity. (Indeed, one can view elliptic curves as “multiplicative tori”, and both the additive and multiplicative tori can be identified as smooth manifolds with the more familiar geometric torus, but we will not use such an identification here.) This fundamental identification with elliptic curves and tori motivates many of the further remarkable properties of elliptic curves; for instance, the fact that tori are obviously an abelian group gives rise to an abelian group law on elliptic curves (and this law can be interpreted as an analogue of the trigonometric sum identities for ). The description of the various meromorphic functions on the torus also helps motivate the more general Riemann-Roch theorem that is a fundamental law governing meromorphic functions on other compact Riemann surfaces (and is discussed further in these 246C notes). So far we have focused on studying a single torus . However, another important mathematical object of study is the space of all such tori, modulo isomorphism; this is a basic example of a moduli space, known as the (classical, level one) modular curve . This curve can be described in a number of ways. On the one hand, it can be viewed as the upper half-plane quotiented out by the discrete group ; on the other hand, by using the -invariant, it can be identified with the complex plane ; alternatively, one can compactify the modular curve and identify this compactification with the Riemann sphere . (This identification, by the way, produces a very short proof of the little and great Picard theorems, which we proved in 246A Notes 4.) Functions on the modular curve (such as the -invariant) can be viewed as -invariant functions on , and include the important class of modular functions; they naturally generalise to the larger class of (weakly) modular forms, which are functions on which transform in a very specific way under -action, and which are ubiquitous throughout mathematics, and particularly in number theory. Basic examples of modular forms include the Eisenstein series, which are also the Laurent coefficients of the Weierstrass elliptic functions . More number theoretic examples of modular forms include (suitable powers of) theta functions , and the modular discriminant . Modular forms are -periodic functions on the half-plane, and hence by Proposition 1 come with Fourier coefficients ; these coefficients often turn out to encode a surprising amount of number-theoretic information; a dramatic example of this is the famous modularity theorem, (a special case of which was) used amongst other things to establish Fermat’s last theorem. Modular forms can be generalised to other discrete groups than (such as congruence groups) and to other domains than the half-plane , leading to the important larger class of automorphic forms, which are of major importance in number theory and representation theory, but which are well outside the scope of this course to discuss.An algebraic (affine) plane curve of degree over some field is a curve of the form
where is some non-constant polynomial of degree . Examples of low-degree plane curves include
- Degree (linear) curves , which are simply the lines;
- Degree (quadric) curves , which (when ) include the classical conic sections (i.e. ellipses, hyperbolae, and parabolae), but also include the reducible example of the union of two lines; and
- Degree (cubic) curves , which include the elliptic curves (with non-zero discriminant , so that the curve is smooth) as examples (ignoring some technicalities when has characteristic two or three), but also include the reducible examples of the union of a line and a conic section, or the union of three lines.
- etc.
Algebraic affine plane curves can also be extended to the projective plane by homogenising the polynomial. For instance, the affine quadric curve would become .
One of the fundamental theorems about algebraic plane curves is Bézout’s theorem, which asserts that if a degree curve and a degree curve have no common component, then they intersect in at most points (and if the underlying field is algebraically closed, one works projectively, and one counts intersections with multiplicity, they intersect in exactly points). Thus, for instance, two distinct lines intersect in at most one point; a line and a conic section intersect in at most two points; two distinct conic sections intersect in at most four points; a line and an elliptic curve intersect in at most three points; two distinct elliptic curves intersect in at most nine points; and so forth. Bézout’s theorem is discussed in this previous post.
From linear algebra we also have the fundamental fact that one can build algebraic curves through various specified points. For instance, for any two points one can find a line passing through the points , because this imposes two linear constraints on three unknowns and is thus guaranteed to have at least one solution. Similarly, given any five points , one can find a quadric curve passing through these five points (though note that if three of these points are collinear, then this curve cannot be a conic thanks to Bézout’s theorem, and is thus necessarily reducible to the union of two lines); given any nine points , one can find a cubic curve going through these nine points; and so forth. This simple observation is one of the foundational building blocks of the polynomial method in combinatorial incidence geometry, discussed in these blog posts.
In the degree case, it is always true that two distinct points determine exactly one line . In higher degree, the situation is a bit more complicated. For instance, five collinear points determine more than one quadric curve, as one can simply take the union of the line containing those five points, together with an arbitrary additional line. Similarly, eight points on a conic section plus one additional point determine more than one cubic curve, as one can take that conic section plus an arbitrary line going through the additional point. However, if one places some “general position” hypotheses on these points, then one can recover uniqueness. For instance, given five points, no three of which are collinear, there can be at most one quadric curve that passes through these points (because these five points cannot lie on the union of two lines, and by Bézout’s theorem they cannot simultaneously lie on two distinct conic sections).
For cubic curves, the situation is more complicated still. Consider for instance two distinct cubic curves and that intersect in precisely nine points (note from Bézout’s theorem that this is an entirely typical situation). Then there is in fact an entire one-parameter family of cubic curves that pass through these points, namely the curves for any (with the convention that the constraint is interpreted as when ).
In fact, these are the only cubics that pass through these nine points, or even through eight of the nine points. More precisely, we have the following useful fact, known as the Cayley-Bacharach theorem:
Proposition 1 (Cayley-Bacharach theorem) Let and be two cubic curves that intersect (over some algebraically closed field ) in precisely nine distinct points . Let be a cubic polynomial that vanishes on eight of these points (say ). Then is a linear combination of , and in particular vanishes on the ninth point .
Proof: (This proof is based off of a text of Husemöller.) We assume for contradiction that there is a cubic polynomial that vanishes on , but is not a linear combination of and .
We first make some observations on the points . No four of these points can be collinear, because then by Bézout’s theorem, and would both have to vanish on this line, contradicting the fact that meet in at most nine points. For similar reasons, no seven of these points can lie on a quadric curve.
One consequence of this is that any five of the determine a unique quadric curve . The existence of the curve follows from linear algebra as discussed previously. If five of the points lie on two different quadric curves , then by Bezout’s theorem, they must share a common line; but this line can contain at most three of the five points, and the other two points determine uniquely the other line that is the component of both and , and the claim follows.
Now suppose that three of the first eight points, say , are collinear, lying on a line . The remaining five points do not lie on , and determine a unique quadric curve by the previous discussion. Let be another point on , and let be a point that does not lie on either or . By linear algebra, one can find a non-trivial linear combination of that vanishes at both and . Then is a cubic polynomial that vanishes on the four collinear points and thus vanishes on , thus the cubic curve defined by consists of and a quadric curve. This curve passes through and thus equals . But then does not lie on either or despite being a vanishing point of , a contradiction. Thus, no three points from are collinear.
In a similar vein, suppose next that six of the first eight points, say , lie on a quadric curve ; as no three points are collinear, this quadric curve cannot be the union of two lines, and is thus a conic section. The remaining two points determine a unique line . Let be another point on , and let be another point that does not lie on either and . As before, we can find a non-trivial cubic that vanishes at both . As vanishes at seven points of a conic section , it must vanish on all of , and so the cubic curve defined by is the union of and a line that passes through and , which must necessarily be . But then this curve does not pass through , a contradiction. Thus no six points in lie on a quadric curve.
Finally, let be the line through the two points , and the quadric curve through the five points ; as before, must be a conic section, and by the preceding paragraphs we see that does not lie on either or . We pick two more points lying on but not on . As before, we can find a non-trivial cubic that vanishes on ; it vanishes on four points on and thus defines a cubic curve that consists of and a quadric curve. The quadric curve passes through and is thus ; but then the curve does not pass through , a contradiction. This contradiction finishes the proof of the proposition.
I recently learned of this proposition and its role in unifying many incidence geometry facts concerning lines, quadric curves, and cubic curves. For instance, we can recover the proof of the classical theorem of Pappus:
Theorem 2 (Pappus’ theorem) Let be two distinct lines, let be distinct points on that do not lie on , and let be distinct points on that do not lie on . Suppose that for , the lines and meet at a point . Then the points are collinear.
Proof: We may assume that are distinct, since the claim is trivial otherwise.
Let be the union of the three lines , , and (the purple lines in the first figure), let be the union of the three lines , , and (the dark blue lines), and let be the union of the three lines , , and (the other three lines). By construction, and are cubic curves with no common component that meet at the nine points . Also, is a cubic curve that passes through the first eight of these points, and thus also passes through the ninth point , by the Cayley-Bacharach theorem. The claim follows (note that cannot lie on or ).
The same argument gives the closely related theorem of Pascal:
Theorem 3 (Pascal’s theorem) Let be distinct points on a conic section . Suppose that for , the lines and meet at a point . Then the points are collinear.
Proof: Repeat the proof of Pappus’ theorem, with taking the place of . (Note that as any line meets in at most two points, the cannot lie on .)
One can view Pappus’s theorem as the degenerate case of Pascal’s theorem, when the conic section degenerates to the union of two lines.
Finally, Proposition 1 gives the associativity of the elliptic curve group law:
Theorem 4 (Associativity of the elliptic curve law) Let be a (projective) elliptic curve, where is the point at infinity on the -axis, and the discriminant is non-zero. Define an addition law on by defining to equal , where is the unique point on collinear with and (if are disjoint) or tangent to (if ), and is the reflection of through the -axis (thus are collinear), with the convention . Then gives the structure of an abelian group with identity and inverse .
Proof: It is clear that is the identity for , is an inverse, and is abelian. The only non-trivial assertion is associativity: . By a perturbation (or Zariski closure) argument, we may assume that we are in the generic case when are all distinct from each other and from . (Here we are implicitly using the smoothness of the elliptic curve, which is guaranteed by the hypothesis that the discriminant is non-zero.)
Let be the union of the three lines , , and (the purple lines), and let be the union of the three lines , , and (the green lines). Observe that and are cubic curves with no common component that meet at the nine distinct points . The cubic curve goes through the first eight of these points, and thus (by Proposition 1) also goes through the ninth point . This implies that the line through and meets in both and , and so these two points must be equal, and so as required.
One can view Pappus’s theorem and Pascal’s theorem as a degeneration of the associativity of the elliptic curve law, when the elliptic curve degenerates to three lines (in the case of Pappus) or the union of one line and one conic section (in the case of Pascal’s theorem).
[This post is authored by Emmanuel Kowalski.]
This post may be seen as complementary to the post “The parity problem in sieve theory“. In addition to a survey of another important sieve technique, it might be interesting as a discussion of some of the foundational issues which were discussed in the comments to that post.
Many readers will certainly have heard already of one form or another of the “large sieve inequality”. The name itself is misleading however, and what is meant by this may be something having very little, if anything, to do with sieves. What I will discuss are genuine sieve situations.
The framework I will describe is explained in the preprint arXiv:math.NT/0610021, and in a forthcoming Cambridge Tract. I started looking at this first to have a common setting for the usual large sieve and a “sieve for Frobenius” I had devised earlier to study some arithmetic properties of families of zeta functions over finite fields. Another version of such a sieve was described by Zywina (“The large sieve and Galois representations”, preprint), and his approach was quite helpful in suggesting more general settings than I had considered at first. The latest generalizations more or less took life naturally when looking at new applications, such as discrete groups.
Unfortunately (maybe), there will be quite a bit of notation involved; hopefully, the illustrations related to the classical case of sieving integers to obtain the primes (or other subsets of integers with special multiplicative features) will clarify the general case, and the “new” examples will motivate readers to find yet more interesting applications of sieves.
On Wednesday, Shou-wu Zhang continued his lecture series. Whereas the first lecture was a general overview of the rational points on curves problem, the second talk focused entirely on the genus 1 case – i.e. the problem of finding rational points on elliptic curves. This is already a very deep and important problem in number theory – for instance, this theory is decisive in Wiles’ proof of Fermat’s last theorem. It was also somewhat more technical than the previous talk, and I had more difficulty following all the details, but in any case here is my attempt to reconstruct the talk from my notes. Once again, the inevitable inaccuracies here are my fault and not Shou-wu’s, and corrections or comments are greatly appreciated.
NB: the talk here seems to be loosely based in part on Shou-wu’s “Current developments in Mathematics” article from 2001.
Recent Comments