Date: Tue, 9 Nov 93 09:40:10 -0600 From: [Permission pending - 1] To: rusin@mp.cs.niu.edu Subject: Re: seeking rational functions as group operations What you want (with F a power series) goes under the name of "Formal groups". You should find many books with this title. In particular there is one by Hazewinkel which is quite comprehensive. One-dimensional formal groups (i.e. F has two variables) are always commutative and they all have a "logarithm" that gives an isomorphism to x+y. The rational functions giving 1-dimensional formal groups have been classified by Coleman and McGuiness but I don't have the reference at hand. [sig deleted -- djr] ============================================================================== Date: Tue, 9 Nov 93 13:14:09 -0800 From: [Permission pending - 2] To: rusin@mp.cs.niu.edu Subject: Re: seeking rational functions as group operations [ a copy has been posted ] There is some work on this done by the people who were working on probabilistic metric spaces because it connects up with trying to get a nice triangle inequality. Much of it is probably only tangentially relevant, but it may be that you will find references to some classical results in these papers. I have B. Schweizer's bibliography on probabilistic metric spaces, but it's from 1977. Anyway, here is what I find. I just have the titles, so it may be that these are irrelevant. The first looks like it should have at least references to the stuff you want, but who can tell from the title? I marked with a * those that seem particularly relevant. Others might be good, too, though. You can search MR under the names of Schweizer, Sklar, Kimberling, and C.H. Ling and see what they did after 1977... * M.J. Frank, On the simultaneous associativity of F(x,y) and x+y-F(x,y) [marked as "To appear" in Aeq. Math., so if you look in Math Reviews around 1977/78/79 you should find it] P. Johnson and A. Sklar, Recurrence and dispersion under iteration of Chebyshev polynomials. J. Math. Anal. and Applic. 54 (1976), 752-771 * M.J. Frank, Associativity in a class of operations on a space of distribution functions. Aeq. Math. 12 (1975) 121-133 [MR 52 #9327] B. Schweizer and A. Sklar, Operations on distribution functions not derivable from operations on random variables, Studia Math. 52 (1974), 43-52 [MR 50 #11377] * C.H. Kimberling, On a class of associative functions, Publ. Math. Debrecen, 20 (1973), 21-39 [MR 48 #11829] T. Erber, B. Schweizer and A. Sklar, Mixing transformations on metric spaces. Commun. Math. Phys. 29 (1973) 311-317 [MR 48 #2343] M.J. Frank, Associative operators on spaces of distribution functions, Ph.D. thesis, Ill. Inst. of Technology (1972) C.H. Kimberling, Associative functions and an application to exchangeable stochastic processes, Ph.D. thesis, Ill. Inst. of Technology (1970) * C.H. Ling, Representation of associative functions, Publicationes Mathematicae, Debrecen, 12, 189-212 [MR 32, #7987] (1965) * C.H. Ling, Representation of associative functions, Ph.D. thesis, Ill. Inst. of Technology (1964) * B. Schweizer and A. Sklar, Associative functions and abstract semigroups, Publicationes Mathematicae, Debrecen, 10, 69-81 [MR 30, #1201] B. Schweizer and A. Sklar, Associative functions and statistical triangle inequalities, Publicationes Mathematicae, Debrecen, 8, 169-186 [MR 24A #2775] ============================================================================== To: David Rusin Subject: Re: seeking rational functions as group operations Date: Wed, 10 Nov 93 13:40:14 -0800 From: "David G. Cantor" In your posting you say: I am looking for examples of rational functions which define an associative binary operation on (most of) a vector space (over the rationals, say). For example on a one-dimensional vector space (Q itself) we have the two polynomials F(x,y)=x+y and F(x,y)=xy which define the group operations on (Q,+) and (Q^*, \times). But it is a typical exercise to show that F(x,y) = x+y+xy is also associative [using x-> x+1 we see that this is the same group as Q^*]. Similarly, F(x,y)=(xy)/(x+y) works almost everywhere [using x -> 1/x we recognize the group (Q,+)] although there is a difficulty when x=-y which I am happy to ignore. . . ------------------------------------------------------------------------ Somethings that come to mind are addition on elliptic curves. There are numerous books in the literature on how to add on such a curve. There are also group laws on the Jacobians o hyperelliptic curves. These are curves given by equations of the form $Y^2=F(X)$ where $F(X)$ is a monic poynomial of degree $2g+1$ ($g$ is called the genusa. I published a paper giving the explicit addition law in Math. of Comp. a few years ago. dgc David G. Cantor Department of Mathematics University of California Los Angeles, CA 90024 dgc@math.ucla.edu ============================================================================== Date: Wed, 10 Nov 1993 16:34:25 -0600 From: David Rusin To: teke@jocosus.matematik.su.se, rusin@math.niu.edu Subject: Re: seeking rational functions as group operations Thanks for your posting! This is just the sort of asnwer I was looking for. However, I am having a bit of trouble reconciling this with the response I got from someone else, reminding me to look at elliptic curves. Viewing such a one as the solutions to the equation y^2=P(x) in Q^2, we can write out the group operation as a pair of (fairly simple) rational functions (which can be viewed as rational maps Q^4 -> Q^2). I am afraid I don't see how to embed such a group into a matrix group as a subgroup; this would seem to suggest there are rational maps Q^4->GLn(Q) for some n which carry the group operation to matrix multiplication. For example, if the elliptic curve is written y^2=x^3+ax+b Then the group inverse is given by (-) (x,y) = (x,-y) and the group operation is (x,y)*(x',y')=(x",y") where set m=(y'-y)/(x'-x) if x<>x', (3x^2+A)/(2y) if x=x' x"=-(x+x')+m^2 y"=-y-m(x"-x) (if x=x' and y=0 then (x,y)*(x',y')=point at infinity=group identity; I know you said the group was supposed to have identity but I think you can substitute 1/x for x to fix that if it is critical.) I would be most curious to see a rational map which carries this to group multiplication. Perhaps the situation is murkier because the associative law only holds on the curve and not on all of Q^2? (that is, ((P0+P1)+P2)-(P0+(P1+P2)) lies in the ideal (y1^2-x1^3-ax1-b, y2^2-x2^3-ax2-b, y3^2-x3^3-ax3-b). Actually it must lie in the intersection of all such ideals as a and b vary. This is a challenging problem to verify with Maple -- whole screenfuls of data go roaring past.) In article you write: >variety part. (I guess I shouldn't have said this as this is exactly the >kind of statement that, according to the PS, was not asked for :-).) Quite all right. dave rusin@math.niu.edu ============================================================================== Date: Wed, 10 Nov 93 16:47:59 CST From: rusin (Dave Rusin) To: dgc@math.ucla.edu Subject: Re: seeking rational functions as group operations Actually there is a funny twist here. I am trying to reconcile your response with a post in sci.math.research about formal groups. I have a hard time visualizing a rational mapping from an elliptic curve to a Lie group which would carry the funny group operation on the curve to ordinary matrix multiplication. I'd like to see it in fact. Possibly none need exist since the group operation on an elliptic curve need not really be a formal group law. Let me clarify: The group operation on a curve E in Q^2 can be written out as a pair of rational functions Q^4 ->Q^2 which are defined for (almost) every point in Q^4, not just those of E. (In fact, these functions carry the pair ((x,y),(x'y')) to the sum of those points in the (usually unique) elliptic curve of the form y^2=x^3+ax+b which contains (x,y) and (x',y').) However, attempting to verify the associative law allows for the possiblity that the three points do not lie on a common elliptic curve, so that applying the rational functions to compute ((P0+P1)+P2)-(P0+(P1+P2)) gives not zero but something (whose numerator is) in the ideal (y0^2-x0^3-ax0-b,y1^2-x1^3-ax1-b,y2^2-x2^3-ax2-b) (Actually it's in this ideal for all a and b, and hence in the intersection of these ideals. The intersection is easy to get by hand, but asking Maple to compute the associative-law difference above shows an emormous formula.) Anyway, the elliptic curves give interesting examples, although the non-associativity places them in a different camp from the usual examples. If there is a rational group homomorphism E--> GLn(Q) I'd like to see it. thanks again, dave rusin@math.niu.edu ============================================================================== Date: Thu, 11 Nov 1993 07:34:26 +0100 From: Torsten Ekedahl To: rusin@mp.cs.niu.edu Subject: Re: seeking rational functions as group operations Careful! In order for you to prove associativity of your suggested group law you have to use the relation y^2=x^3+ax+b which means that x and y may not be assumed to be algebraically independent. In other words the group manifold you describe is 1-dimensional and if you want to make it equivalent to a rational group law it also should be 1-dimensional. However, making it equivalent to a rational group law would in particular mean that the function field k(x)[y]/(y^2-x^3-ax-b) would be isomorphic to k(t) and this is known to be false. In fact, if we stick to an algebraically closed field then the assumption that we have a rational group law is precisely what excludes things like elliptic curves and forces the algebraic group to be a matrix group. On the other hand, if you allow power series as transformations S then the elliptic curve group law is equivalent to F(x,y)=x+y. > However, I am having a bit of trouble reconciling this with the > response I got from someone else, reminding me to look at elliptic curves. There are no contradictions in mathematics! :-)