Talk:Complex multiplication

I am expecting multiplication in the complex field at this(Complex multiplication) title. --MarSch 12:52, 3 Jun 2005 (UTC)

How's that? Dmharvey Talk 16:57, 3 Jun 2005 (UTC)

better than nothing --MarSch 18:31, 3 Jun 2005 (UTC)

Apparently there are many n for which $e^{\pi {\sqrt {n}}}$ is very near an integer, for example n = 58 and 652. How come? - Fredrik | talk 21:39, 16 September 2005 (UTC)

I'm attempting to figure that out right now... I have a very good understanding of undergraduate mathematics, and a beginning understanding of set theory, group theory, analysis, et cetera. I'm finding this article somewhat difficult to follow, but I really, really want to know the answer to the question you asked, so I'm going to try. --Monguin61 04:12, 15 December 2005 (UTC)

If "undergraduate mathematics" does not include at least "set theory, group theory, analysis, et cetera", I'm a bit curious what it does include... Michael Lee Baker (talk) 04:39, 29 July 2015 (UTC)

Trying to follow explanation

I guess this is evident to anyone who knows what theyre doing, so maybe the article doesn't need the notation explained, but I was wondering if anyone could tell me what

$\mathbf {Z} \left[{\frac {1+{\sqrt {-163}}}{2}}\right]$

actually means? I assume the Z denotes a group of some sort, and the stuff inside the brackets is a parameter defining the group, but what type of group exactly? Or am I totally off? --Monguin61 04:24, 15 December 2005 (UTC)

Z[α] is the subring of the complex numbers generated by the complex number α if you want it in simple terms. Charles Matthews 08:53, 15 December 2005 (UTC)

Z is the set of all integers. The thing in square brackets is something not belonging to Z, being "adjoined", so that we get the smallest set contaning all integers and containing the thing in square brackets and closed under addition, subtraction, and multiplication. See Herstein's advance undergraduate text Topics in Algebra. Michael Hardy 00:15, 16 December 2005 (UTC)

I will add a link explaining this notation. But I cannot either follow the "explanation", I mean, some more details (even if unprecise and handwaving), or at least an "external link" seem necessary, because readers come from "far away" to this page (referred to in pi and other places), hungry for this explanation (why dist( exp(...), Z) is so small )). — MFH:Talk 14:14, 16 March 2006 (UTC)

Trying to follow ...

I followed this article up to this point:

This is a typical elliptic curve with complex multiplication, in the sense that over the complex number field they are all found as such quotients, in which some order in the ring of integers in an imaginary quadratic field takes the place of the Gaussian integers.

Is this trying to say that every elliptic curve can be obtained as C/cZ[F] where c is some complex number, and F is an imaginary quadratic field? Is this furthermore implying that such a Z[F] is a unique factorization domain? Last but not least, is there an implication that for any f in F, that exp (f\pi) == close approximation to integer?

Anyway, I barely understand what I just asked; this is all new ot me. I'll try to hunt down book references for this topic, but if anyone has one handy, let me know? linas 01:56, 19 January 2006 (UTC)

Serge Lang, Elliptic Functions might suit you. Every elliptic curve is C/Λ for a lattice Λ, but the cases where Λ has anything to do with a quadratic field are very special. The associated τ must be one of a countable set of algebraic numbers, for example. Charles Matthews 08:50, 19 January 2006 (UTC)

OK, I understand the bit about elliptic curves; I found something on the web that skteches the proof, viz. along the lines that the j-invariant is an algebraic integer. If the ring of integers is a UFD, then j is a "rational integer". By then truncating the fourier expansion for j, (the q-series expansion for j) (justified because because q is so small), one obtains the Ramanujan number \exp \pi \sqrt{163}.

Now for the trick question: are there similar results for an "irrational" quadratic "field" (well, ok, ring, then) of the form

$K=\mathbb {Q} [{\sqrt {\pi n}}]$

I have a near-identity that clearly has a square-root of pi sitting next to an integer, and am trying to explain this ident. I'm looking for anything related to such a field... linas 18:38, 20 January 2006 (UTC)

Simple answer: no. The ring is just a polynomial ring, and it doesn't embed in the complex plane as a lattice. Charles Matthews 19:31, 20 January 2006 (UTC)

Since sqrt(pi) is not algebraic, I would even dare to say that this is isomorphic to Q[X]. — MFH:Talk 02:01, 1 April 2006 (UTC)

correction

The equation α²=α-81 should be α²=α-41 on this page —Preceding unsigned comment added by 192.91.173.36 (talk) 01:47, 7 February 2010 (UTC)

Abrupt introduction of α

In the "Sample consequence" section, it says "Here α² = α − 41" with no prior occurrence of α. Is something missing? Wideangle (talk) 19:17, 30 March 2010 (UTC)

I wondered that too. Looking at one of the comments above ("Z[α] is the subring of the complex numbers generated by the complex number α"), I suspect that this is standard terminology for experts in the field. But it certainly doesn't seem clear enough for a non expert.

I've added a fragment to clarify α, but it needs more expert intention to clarify the meaning and import of the whole of this paragraph - there is no mention elsewhere in the article of S[α] or "the required polynomials" (whatever they might be). Hv (talk) 07:08, 23 October 2010 (UTC)

Article's name should be changed

To reduce confusion, the name of this article should be changed. I suggest it be changed to either "Complex multiplication (elliptic curves)" or "Complex multiplication (elliptic functions)" Wideangle (talk) 00:22, 31 March 2010 (UTC)

No, there is no requirement to disambiguate in that fashion: adding things in parentheses, except where there is another article of a related name, is against general title conventions. Charles Matthews (talk) 14:06, 31 March 2010 (UTC)

I'm not saying that there is a requirement to make such a change. I am suggesting that the change be made in order to reduce confusion. The vast majority of people who think they know what "complex multiplication" is think that it means the multiplication of complex numbers (which makes a lot of sense, by the way). My background is: BA in Mathematics from the University of California at Berkeley. And yet, I never encountered the "elliptic curves" meaning of "complex multiplication" until three days ago when I came upon a link to this article in the John Tate article. Wideangle (talk) 00:52, 3 April 2010 (UTC)

There is a hatnote explaining that the article is not about multiplication of complex numbers. Looking at Wikipedia:Article titles, the title seems to conform to the points Recognizable (commonly used in reliable sources), Easy to find (names and terms that readers are most likely to look for in order to find the article), Precise, and Concise. Basically there is nothing in that guideline to justify a change. Charles Matthews (talk) 08:39, 3 April 2010 (UTC)

Regardless of what the article's title is, I think this point should be made clearer -- It's surprising easy to overlook the note (I did), and I think a good many readers will be expecting information about multiplying complex numbers. KLuwak (talk) 15:49, 28 February 2011 (UTC)

First section of article

The first section of the article is quite rough and could benefit from a rewrite ("ring of analytic automorphic group" doesn't make any sense; "ring of automorphisms" should read "group of automorphisms"; $\mathbf {C} ^{3}$ is incorrectly used when we should be speaking of the projective plane; grammar is generally poor and so on). I may eventually get around to it. Michael Lee Baker (talk) 04:33, 29 July 2015 (UTC)

Article needs a definition of its subject

In fact, a definition of its subject is the only thing that every encyclopedia article must have. Yet this article lacks such a definition.

(The article does say the following:

"Consider an imaginary quadratic field $K=\mathbb {Q} ({\sqrt {-d}})\ ,d\in \mathbb {Z} ,d>0$. An elliptic function $f$ is said to have complex multiplication if there is an algebraic relation between $f(z)$ and $f(\lambda z)$ for all $\lambda$ in $K$ ."

The concept of an "algebraic relation" between two functions f(z) and g(z) would normally mean that there is a polynomial P(X,Y) in two variables such that for all values of z that make sense, we have P(f(z), g(z)) = 0.

But in this article we are not told whether the coefficients of P(X,Y) are merely complex numbers, or if they are required to belong to the field K.

Or am I missing something?2600:1700:E1C0:F340:ED3B:6DA2:E8FC:C260 (talk) 00:22, 24 May 2019 (UTC)

Finite field

I don't understand why an elliptic curve over a finite field has an endomorphism ring larger than the integers, due to the Frobenius endomorphism. What is it I do not understand? The Frobenius endomorphism is an endomorphism on a comnutative ring with characteristic p. Hence the refered Frobenius endomorphism cannot be an endomorphism on the elliptic curve itself. But if it has to be an endomorphism on the ring of endomorphisms - what makes sense, but this ring is not abelian in general. If the field is a prime field with charateristic p, a point on the curve has coordinates x and y, and the Frobenius endomorphism, defined for the elliptic curve, maps x -> x^p=x, and y -> y^p=y, hence it is the identity. Madyno (talk) 22:21, 13 December 2021 (UTC)

Intro

From ghe introduction: CM means the ring of endomorphism is larger than $\mathbb {Z}$. This may raise the idea the ring should contain the integral numbers or should be infinite. In my opinion the correct formulation would be there is a non-trivial endomorphism, that is an endomorphism not induced by the integers. Madyno (talk) 23:15, 7 December 2021 (UTC)