Taniyama Shimura Weil Argument

For the context - see the ultra high-level overview of the statement on TaniyamaShimuraWeilTheorem. Now read on for as long as you can care.

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I don't dare touch this because I was one of the parties involved, but I think it's past time someone pulled the facts from this and removed the arguments.

[first pass editing (2005-Dec-14): simple deletion of pure ThreadMode ad hominems and otherwise completely stale side-comments lacking technical content, without touching comments with technical content. The remaining text is choppy and unclear. So was the original; that is not an artifact of deletion of non-technical content. The remaining text is unclear about technical content; so was the original. This still needs quite a bit of work.]

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Note

Even though elliptic curves aren't actually ellipses, the name isn't crazy. They arise in questions concerning ellipses, such as lengths of arcs and so on, and can also be considered to be generalizations of ellipses in the sense that they are multiply periodic functions on toroids, which generalizes the notion of singly periodic functions like ellipses.

[Isn't that [roughly] true of elliptic functions rather than elliptic curves?]

• Yes and no. The elliptic curve solutions occur on the toroid of its characteristic elliptic function, which interconverts the two; it's all pretty much one topic. More importantly, for this page it's just a side topic anyway. It leads immediately into interesting issues such as bridging the gap between trigonometric periodicity and number theory periodicity, which is particularly interesting and of rather critical importance, but...
  • [What do "solutions occur on" and "number theory periodicity" mean? Also, why do you perceive a gap, given that periodicity arises in both areas?]
  • "solutions occur on": http://mathworld.wolfram.com/EllipticCurve.html
    • That site uses "confined to a region of space" instead of "occurs on", but doesn't explain what is meant.
  • "number theory periodicity": finite cyclic group with representation N mod prime.
  • "why do I perceive a gap": I don't "perceive" one, it's just that, as is well known, trig periodicity and cyclic group periodicity is not the same thing, they are analogous, and under certain formalisms, isomorphic. I suspect you already know all of this; otherwise you'd be in the position of claiming that periodicity 2pi numerically equals the order of the cyclic group of order 7. More to the point, the discrete branches of mathematics and the continuous branches of mathematics are largely separate, but with recurring connections, and this is one of the important areas of connections. If you would like this to be phrased in a better way, be my guest.
  • "given that periodicity arises in both areas": that was my point.
    • Some functions are periodic. Functions (some periodic, some not) are used in both areas. Likewise for symmetry, but one wouldn't talk of symmetry as bridging the gap between the areas. There doesn't seem to be any significant point to make here.

[The latter was disagreed with but without actual argument, in the original]

Isn't it being unnecessarily obscure to leap from "Elliptic curves are equations in two variables" to "they are multiply periodic functions on toroids"? (Equations and functions are different, and the meaning of "on toroids" is unclear (Google finds nothing for "functions on toroids").) Also, assuming "Proofs are starting to filter out" means that some corresponding urls exist, how about giving some examples of them? I tried Google without success.

[You said "equations and functions are different" - maybe they are, and maybe they're not, but why don't you explain how you think they're different, and why that is an issue here, and see if we can straighten that out? Certainly - an equation states that two expressions are equal, whereas a function maps one domain to another. That means that they're different, and gives no indication of what it would mean to say that a function is "defined on a toroid". As you suggest, MathIsHard, so it's unhelpful to make it even harder by using unclear, inconsistent or misleading terminology.]

[...non-technical crud deleted...]

Firstly, ellipses, and any other geometrical object, and most mathematical objects in general, can and ought to be thought of in more than one way. To consider an ellipse simply as a specific two dimensional symmetric closed curve is to miss the point (so to speak). In some very real sense an ellipse *is* a function - it is (y/a)^2 + (x/b)^2 = 1. Thinking of it this way means that you can start to apply algebra and get the cross-fertilization of the two areas. It was an insight very similar to this that gave the connection between FLT and elliptic curves. Specifically, Frey noticed that is there is a non-trivial solution to Fermat's equation, so A^n+B^n=C^n, then the elliptic curve y^2=x(x-a^n)(x+b^n) has some extreme characteristics. That observation led to Wiles' solution.

The original questioner said (including their modifications):

Isn't it being unnecessarily obscure to leap from "Elliptic curves are equations in two variables" to "they are multiply periodic functions on toroids"? (Equations and functions are different, and the meaning of "on toroids" is unclear (Google finds nothing for "functions on toroids").) Also, assuming "Proofs are starting to filter out" means that some corresponding urls exist, how about giving some examples of them? I tried Google without success.

No, it's not being unnecessarily obscure. Although they are viewed from different points of view, equations and functions are not fundamentally different. Consider the equation x^3+y^2=2. This can be regarded as an equation, yes, but it is also a function. In fact, it is more than one function. We have y = (x^3-2)^(1/2) and x = (y^2-2)^(1/3). You may object and say that these aren't functions because they have more than one value. If so, I recommend you read http://mathworld.wolfram.com/RiemannSurface.html to see why we can usefully modify the apparent domain to make a multi-valued function (i.e. "relation") into a function.

To talk about a "function on a toroid" is simply to say that the function's domain is the toroid - we are assigning values to each location on the toroid. We might assign multiple values, and that leads to the idea of having multiple copies of the toroid which are cut and glued to ensure continuity of the function. To say that a function is multiply periodic is to say that it's a bit like a sine wave, only its domain is closed and finite.

I personally don't understand the original assertion about elliptic curves being generalizations of ellipses. It seems to say:

• Ellipses can be thought of as singly periodic functions
• Elliptic curves can be thought of as multiply periodic functions on toroids

Perhaps someone could explain both of those remarks.

• I'm not sure what is confusing, that needs clarification. It's stronger than "can be thought of"; a singly periodic quadratic polynomial in N variables is an ellipse/ellipsoid. A doubly-periodic analytic function is an elliptic function. (Both definitions need very mild tightening up to avoid nitpicks, but such does not change the idea.) Double-periodicity obviously generalizes single-periodicity, and analytic functions obviously generalize polynomials (in many senses, but starting with the fact that they are polynomials of infinite degree). Also, doubly-periodic clearly implies genus 1, and in the right setting, vice-versa, thus, toroidal.
  • The reference was to elliptic curves, not elliptic functions - the two can be connected, but are not the same. Elliptic functions are doubly-periodic, but that doesn't make elliptic curves doubly-periodic.
  • What is the period in the case of the ellipse (y/a)^2 + (x/b)^2 = 1?
    • [...] the period is 2pi. The important properties of the system are invariants, they are coordinate-free. You are asking a coordinate-based question, and predictably will be unhappy that the answer is in a different coordinate system. Insight is to be found by examining system properties in a non-metric topological space where distracting issues of coordinate system do not interfere. -- Doug
      • A function's periodicity is always coordinate-based. An arbitrary change of coordinates is not allowed. To see that, suppose x and y are real and y=f(x), so that (in polar coordinates r, theta) r*sin(theta)=f(r*cos(theta)). Now r seems to be a periodic function of theta, with period 2pi (since sin(theta) and cos(theta) have a common period of 2pi), but f was an arbitrary function which needn't have any special properties. One can't even swap x and y. Arcsin(x) is not periodic, but its inverse, sin(x), is periodic. The ellipse mentioned above certainly has important properties, but periodicity is not one of them.
      • Functions can be specified in a coordinate free manner. In fundamental physics, this is even the preferred method.
      • In such cases, one can choose an alternative wording to "coordinate-based", but my points still apply. Avoiding the word "coordinate" doesn't alter the fact that any function, by definition, is a map from its domain to its co-domain.
      • Take a good close look at a circle. The fact that it is CIRCULAR does not change when you decide to go back and forth between Cartesian and polar coordinates. It doesn't even waver. It just sits there being circular. Note how the line on its boundary has no beginning and no end. It is a closed loop. It is PERIODIC because it repeats, no matter where on the boundary you start. That is a QUALITATIVE PROPERTY that does not change under coordinate transformation, although as I said originally, the quantitative value of such things is obviously coordinate dependent. A circle is periodic, there is no ambiguity there.
      • With a circle, there are two things involved. First, it's a finite closed loop, so if you proceed along the loop, you eventually return to your starting point. That property holds for every such loop and isn't coordinate-based. However, it's not called periodicity. The second property is that any rotation (in the plane of the circle) about its center moves each point of the circle to a corresponding point on the circle - a point at the same distance from the center. That is indeed periodicity, but only if the distance from the centre corresponds to the function value, not in Cartesian coordinates. Compare the situation with a straight line. Every straight line maps to itself under any translation in the direction of the line. But the function y=m*x+c is not periodic unless m=0 (i.e., unless the coordinate system is specially chosen).

      • I'm not incorrect. You are simply not recognizing the standard meaning of the term "periodic" as applied to a function. When it can correctly be applied in relation to specially chosen coordinate systems, it does not follow that it can correctly be applied regardless of the coordinate system. Use a different term if that's the meaning you want.
      • [[Hey, don't you realize, at an adequately abstract level this really is periodic! Can't you think abstractly?]]

      • "periodic function: If f is a (non-constant) function from R into R and there exists a number q such that, for all x that belong to R, f(x+q) = f(x), then we say that f is a periodic function. The period of a periodic function is defined to be the smallest positive number p, if any, satisfying {forall x belonging to R: f(x + p) = f(x)}." (emphasis in original; A Handbook of Terms Used in Algebra and Analysis, A.G. Howson, Cambridge University Press, c. 1972, ISBN 0521096952 )
      • "If f(x), defined on a linear space X, satisfies the relation f(x + w) = f(x) for some w belonging to X and all x belonging to X, the number w is called a period of f(x), and f(x) with a period other than zero is called a periodic function. The set P of all periods of f(x) forms an additive group contained in X. If a basis w_1, ..., w_n of the additive group P exists, its members are called fundamental periods of f(x). Any continuous, nonconstant, periodic function of a real variable has only one positive fundamental period and is called a simply periodic function. The trigonometric functions are typical examples; sin x and cos x have the fundamental period 2pi; tan x and cot x have the fundamental period pi."
      • "A single-valued nonconstant meromorphic function of n complex variables cannot have more than 2n fundamental periods that are linearly independent on the real number field. A function of one complex variable with two fundamental periods is called a doubly periodic function."
      • "A doubly periodic function f(u) meromorphic on the complex plane is called an elliptic function. (emphasis in original; section 144 B "Elliptic Functions" in the Encyclopedic Dictionary of Mathematics volume 1, by the Mathematical Society of Japan, ed. S. Iyanaga and Y. Kawada, MIT Press, first MIT Press paperback edition, c. 1977, 1980, ISBN 0262590107 )
      • That firmly settles the precise definition (which is not all that different than the non-technical meaning of "periodic"). An ellipse obviously satisfies that definition. It is equally obvious that, although an ellipse always satisfies that, and thus is always periodic, regardless of how you choose to define the ellipse, its period on the other hand is coordinate-dependent. As I've been saying.
      • [[An ellipse is not even a function, let alone a periodic function. For simplicity, let's consider the very simple ellipse {(x,y)| x**2 + y**2 = 1}. This cannot be a function since a function is a set of ordered pairs, where (x,y) in F and (x,z) in F implies y=z. This is not true of this ellipse, or any ellipse, or any closed curve in the plane. Hence an ellipse is not a function, as the term is used in mathematics, and a fortiori it is not a periodic function. "Elliptic curves" are not "ellipses" so the discussion of "elliptic curves" is irrelevant to the question of whether an ellipse is a periodic function.]]
      • [[Now what you can do, is make this ellipse the range of a function, and furthermore make a function that describes a path which continuously runs through the ellipse over and over again. A function such as lambda x:R.(sin(x), cos(x)) is an example. Now you can discuss the period of such a function. The example has a period of 2pi, and it is a periodic function as a result. But the ellipse which is its range does not have a period. In fact, as the other poster said, you cannot assign 2pi or anything else to it as a period, since it is also the range of many other path functions such as lambda x.R:(sin(x/2),cos(x/2)) which has a different (minimal) period, and also the range of path functions which have no period at all such as lambda x.R:(sin(2**x),cos(2**x)) and are thus not periodic.]]

      • For simplicity, let us consider the polar equation of a circle, r = a. (http://mathworld.wolfram.com/Circle.html). This is a simple function, not a relation, and is not multivalued. The fact that there is a way of expressing ellipses as relations rather than functions does not mean they are inherently such, that's just one expression, and not the simplest. They can also be specified as parametric functions. Or as the function that solves a differential geometry equation. Or in many ways. If one uses one of those ways in which they are indeed functions, then they are periodic functions. When expressed in the complex plane as the sum of a real sine and imaginary cosine, the periodicity is perhaps especially clear. The cartesian relation is actually equivalent, it's just messier since you have to deal with branches of a multivalued relation, but it doesn't change the nature of the circle to take that approach or any of the others.
      • [[No, sorry, I am in agreement with the other guy. An ellipse is not a function, sorry, it is a set of points that cannot be a function by its very nature, as already explained. To correct your most current confusion, an equation is not a function either, it is a constraint that may define a set of points that may or may not be a function. In the case of the equation of a circle, it defines a set of points, but that set of points is not a function, a fortiori not a periodic function. As I said, a path that runs through the circle may very well be a periodic function (although it need not be) but this does not make the circle a periodic function]]
        • You seem to have a rather limited view of what things are and are not. Many advances in mathematics have been achieved by realizing that something that is obviously one thing can, if looked at in the right way, also be considered something else. The TSW theorem is exactly a case in point. To some extent, and in some sense, elliptic curves can be considered to be modular forms in disguise, and vice versa. Insisting, for example, that an ellipse is a collection of points in a plane and nothing else is evidence that you haven't understood this point that I thought I had made repeatedly. Perhaps you don't agree with it, in which case it would be worth hearing your case for the claim that there is no utility in considering this point of view.
        • Different views are alternatives, not explanations to be introduced in arrears to justify confusing statements. No theorem about functions requires you to redefine basic terms (such as function) before you can understand what the theorem is about.
      • I agree that an equation certainly can be a constraint that defines something that may or may not be a function. In the case that it happens to, I see no problem in saying that the equation defines a function, but that seems just a terminology issue -- note that similarly, mathematicians quite frequently talk about "multivalued functions", when strictly you'd want them to say "relations" -- because qualifying it with "multivalued" means there's no danger of confusion.
      • More importantly, why would you claim that the polar equation I mentioned, r = a, does not ...ahhhh, that was it. Ok, let us talk about the function defined by the polar equational constraint r = a. It is not multi-valued. It's a clean map. The functional mapping thus specified, I claim to be periodic. "Path" comes up with a parametric approach, or with a definition by path of integration, but doesn't come up in every single approach, so I don't think we have to go there.
      • [[We are talking about a plane, in which various objects exist: note that your source says "A circle is the set of points in a plane that are equidistant from a given point O.". Normally planes are parameterized in a regular Cartesian sense. Yes, you could I suppose parameterize the plane in polar coordinates, but once you do so, the only circles which will be functions are those close enough to the origin. It is just a fluke of the parameterization that will not work for all circles or more general closed curves in that plane. So most circles are still not functions even in such a system. Moreover, the few circles that are functions are still not periodic functions since in polar coordinates, the entire plane is nothing but the points {(r,theta) | r in R, theta in [0, 2pi[ }.]]
        • "It's a clean map?" That sounds more like a song lyric than mathematics. Anyway, there is some function whose graph, using polar coordinates, is the circle provided the origin isn't outside the circle, and regardless of whether the circle is close to the origin (on the other hand, if the origin is in the interior of the circle, the circle really is in some sense close to the origin...) The same holds true for any "convex" loop. For the circle, the function isn't periodic except in the special case where the origin is the center of the circle. For the ellipse, the same is true, but the period is pi, not 2pi. That is because the ellipse is symmetrical about two axes, and periodicity is also possible for other closed loops, such as a regular polygon. The common property is more usually called "rotational symmetry", but that term still implies the same polar coordinates, with the origin being both the center of rotation and the center of the closed loop.
• Nor is this purely retroactive analysis; the historical development of elliptic functions consciously created analogs of trigonometric notions and functions. -- Doug

To say "it is more than one function" is to admit it is not a function. The equation x^3+y^2=2 describes a relation, which you effectively concede would be the better term. However, there is a huge gulf between "equations in two variables" and "multiply periodic functions on toroids". Suppose we convert y^2 = a*x^3 + b*x^2 + c*x + d to y = (a*x^3 + b*x^2 + c*x + d)^(1/2) or -(a*x^3 + b*x^2 + c*x + d)^(1/2) and consider this as a (multi-valued) function of a complex variable. The domain of the function is the complex plane, or possibly some part of the complex plane, neither of which is a toroid. Yet it is stated both that elliptic curves are equations of the kind I've just mentioned and that they are multiply periodic functions on toroids. Also, it's claimed that they are surfaces like a torus but in a four-dimensional space. There seems to be some confusion. A function defined in terms of a complex variable must have the complex plane (or part of it) as its domain, not a toroid. Also, the function mentioned doesn't appear to be periodic. Ellipses are also referred to as periodic functions, but even when ellipses are interpreted as multivalued functions, those functions are certainly not periodic. Maybe number theorists have there own definition of periodic which needs to be explained here to make sense of this. Whatever the definition, it cannot be as vague as "a bit like a sine wave". The sine function is genuinely periodic (since sin(x) = sin(x + pi) for all x). Such periodicity has nothing to do with whether the domain of the function is closed or finite.

Actually, sin(x) = -sin(x + pi) for all x. Sine has fundamental period 2pi.

Another oddity is that elliptic curves and modular forms are both described as complex functions, yet they are also said to be "from totally different areas of mathematics". See http://log24.com/log03/1130.htm for some criticism of that assertion.

• Elliptic curves are equations of the form y^2=cubic_in_x where x and y are complex numbers. Modular forms are functions from the complex upper half plane to the complex upper half plane satisfying some intricate and esoteric conditions. They are totally different objects. The claims on that URL appear little different from thousands of missives received by almost every major department of mathematics over the decades since the prize for a proof of FLT was announced. To an admittedly brief glance it looks like flying saucer material.
• The author of that (log24.com) site himself notes that his site is listed on a web-cranks page, so whether he is right or wrong, he clearly is aware that his view is not mainstream, at minimum.
  • However, unlike cranks, he isn't presenting his own mathematical arguments, but seems to be simply quoting selectively from reputable sources. He does so rather emphatically and arrogantly, but that doesn't mean he's got it all wrong.