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the taxicab curve

(After-math of last week’s second year lecture on elliptic
curves.)

We all know the story of Ramanujan and the taxicab, immortalized by Hardy

“I remember once going to see him when he was lying ill at Putney. I had ridden in taxicab no. 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it’s a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways’.”

When I was ten, I wanted to become an archeologist and even today I can get pretty worked-up about historical facts. So, when I was re-telling this story last week I just had to find out things like :

the type of taxicab and how numbers were displayed on them and, related to this, exactly when and where did this happen, etc. etc. Half an hour free-surfing further I know a bit more than I wanted.

Let’s start with the date of this taxicab-ride, even the year changes from source to source, from 1917 in the dullness of 1729 (arguing that Hardy could never have made this claim as 1729 is among other things the third Carmichael Number, i.e., a pseudoprime relative to EVERY base) to ‘late in WW-1’ here

Between 1917 and his return to India on march 13th 1919, Ramanujan was in and out a number of hospitals and nursing homes. Here’s an attempt to summarize these dates&places (based on the excellent paper Ramanujan’s Illness by D.A.B. Young).

(may 1917 -september 20th 1917) : Nursing Hostel, Thompson’s Lane in Cambridge.
(first 2 a 3 weeks of october 1917) : Mendip Hills Senatorium, near Wells in Somerset. (november 1917) : Matlock House Senatorium atMatlock in Derbyshire.
(june 1918 – november 1918) : Fitzroy House, a hospital in Fitzroy square in central London. (december 1918 – march 1919) : Colinette House, a private nursing home in Putney, south-west London. So, “he was lying ill at Putney” must have meant that Ramanujan was at Colinette House which was located 2, Colinette Road and a quick look with Google Earth

shows that the The British Society for the History of Mathematics Gazetteer is correct in asserting that “The house is no longer used as a nursing home and its name has vanished” as well as.”

“It was in 1919 (possibly January), when Hardy made the famous visit in the taxicab numbered 1729.”

Hence, we are looking for a London-cab early 1919. Fortunately, the London Vintage Taxi Association has a website including a taxi history page.

“At the outbreak of the First World War there was just one make available to buy, the Unic. The First World War devastated the taxi trade.
Production of the Unic ceased for the duration as the company turned to producing munitions. The majority of younger cabmen were called up to fight and those that remained had to drive worn-out cabs.
By 1918 these remnant vehicles were sold at highly inflated prices, often beyond the pockets of the returning servicemen, and the trade deteriorated.”

As the first post-war taxicab type was introduced in 1919 (which became known as the ‘Rolls-Royce of cabs’) more than likely the taxicab Hardy took was a Unic,

and the number 1729 was not a taxicab-number but part of its license plate. I still dont know whether there actually was a 1729-taxicab around at the time, but let us return to mathematics.

Clearly, my purpose to re-tell the story in class was to illustrate the use of addition on an elliptic curve as a mean to construct more rational solutions to the equation $x^3+y^3 = 1729 $ starting from the Ramanujan-points (the two solutions he was referring to) : P=(1,12) and Q=(9,10). Because the symmetry between x and y, the (real part of) curve looks like

and if we take 0 to be the point at infinity corresponding to the asymptotic line, the negative of a point is just reflexion along the main diagonal. The geometric picture of addition of points on the curve is then summarized
in

and sure enough we found the points $P+Q=(\frac{453}{26},-\frac{397}{26})$ and $(\frac{2472830}{187953},-\frac{1538423}{187953}) $ and so on by hand, but afterwards I had the nagging feeling that a lot more could have been said about this example. Oh, if Im allowed another historical side remark :

I learned of this example from the excellent book by Alf Van der Poorten Notes on Fermat’s last theorem page 56-57.

Alf acknowledges that he borrowed this material from a lecture by Frits Beukers ‘Oefeningen rond Fermat’ at the National Fermat Day in Utrecht, November 6th 1993.

Perhaps a more accurate reference might be the paper Taxicabs and sums of two cubes by Joseph Silverman which appeared in the april 1993 issue of The American Mathematical Monthly.

The above drawings and some material to follow is taken from that paper (which I didnt know last week). I could have proved that the Ramanujan points (and their reflexions) are the ONLY integer points on $x^3+y^3=1729 $.

In fact, Silverman gives a nice argument that there can only be finitely many integer points on any curve $x^3+y^3=A $ with $A \in \mathbb{Z} $ using the decomposition $x^3+y^3=(x+y)(x^2-xy+y^2) $.

So, take any factorization A=B.C and let $B=x+y $ and $C=x^2-xy+y^2 $, then substituting $y=B-x $ in the second one obtains that x must be an integer solution to the equation $3x^2-3Bx+(B^2-C)=0 $.

Hence, any of the finite number of factorizations of A gives at most two x-values (each giving one y-value). Checking this for A=1729=7.13.19 one observes that the only possibilities giving a square discriminant of the quadratic equation are those where $B=13, C=133 $ and $B=19, C=91 $ leading exactly to the Ramanujan points and their reflexions!

Sure, I mentioned in class the Mordell-Weil theorem stating that the group of rational solutions of an elliptic curve is always finitely generated, but wouldnt it be fun to determine the actual group in this example?

Surely, someone must have worked this out. Indeed, I did find a posting to sci.math.numberthy by Robert L. Ward : (in fact, there is a nice page on elliptic curves made from clippings to this newsgroup).

The Mordell-Weil group of the taxicab-curve is isomorphic to $\mathbb{Z} \oplus \mathbb{Z} $ and the only difference with Robert Wards posting was that I found besides his generator

$P=(273,409) $ (corresponding to the Ramanujan point (9,10)) as a second generator the point
$Q=(1729,71753) $ (note again the appearance of 1729…) corresponding to the rational solution $( -\frac{37}{3},\frac{46}{3}) $ on the taxicab-curve.

Clearly, there are several sets of generators (in fact that’s what $GL_2(\mathbb{Z}) $ is all about) and as our first generators were the same all I needed to see was that the point corresponding to the second Ramanujan point (399,6583) was of the form $\pm Q + a P $ for some integer a. Points and their addition is also easy to do with sage :

sage: P=T([273,409])
sage: Q=T([1729,71753])
sage: -P-Q
(399 : 6583 : 1)

and we see that the second Ramanujan point is indeed of the required form!

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the noncommutative manifold of a Riemann surface

The
natural habitat of this lesson is a bit further down the course, but it
was called into existence by a comment/question by
Kea

I don’t yet quite see where the nc
manifolds are, but I guess that’s coming.

As
I’m enjoying telling about all sorts of sources of finite dimensional
representations of $SL_2(\mathbb{Z}) $ (and will carry on doing so for
some time), more people may begin to wonder where I’m heading. For this
reason I’ll do a couple of very elementary posts on simple examples of
noncommutative manifolds.

I realize it is ‘bon ton’ these days
to say that noncommutative manifolds are virtual objects associated to
noncommutative algebras and that the calculation of certain invariants
of these algebras gives insight into the topology and/or geometry of
these non-existent spaces. My own attitude to noncommutative geometry is
different : to me, noncommutative manifolds are genuine sets of points
equipped with a topology and other structures which I can use as a
mnemotechnic device to solve the problem of interest to me which is the
classification of all finite dimensional representations of a smooth
noncommutative algebra.

Hence, when I speak of the
‘noncommutative manifold of $SL_2(\mathbb{Z}) $’ Im after an object
containing enough information to allow me (at least in principle) to
classify the isomorphism classes of all finite dimensional
$SL_2(\mathbb{Z}) $-representations. The whole point of this course is
to show that such an object exists and that we can make explicit
calculations with it. But I’m running far ahead. Let us start with
an elementary question :

Riemann surfaces are examples of
noncommutative manifolds, so what is the noncommutative picture of
them?


I’ve browsed the Google-pictures a bit and a picture
coming close to my mental image of the noncommutative manifold of a
Riemann surface locally looks like the picture on the left. Here, the checkerboard-surface is part of the Riemann surface
and the extra structure consists in putting in each point of the Riemann
surface a sphere, reflecting the local structure of the Riemann surface
near the point. In fact, my picture is slightly different : I want to
draw a loop in each point of the Riemann surface, but Ill explain why
the two pictures are equivalent and why they present a solution to the
problem of classifying all finite dimensional representations of the
Riemann surface. After all why do we draw and study Riemann
surfaces? Because we are interested in the solutions to equations. For
example, the points of the _Kleinian quartic Riemann
surface_ give us all solutions tex \in
\mathbb{C}^3 $ to the equation $X^3Y+Y^3Z+Z^3X=0 $. If (a,b,c) is such
a solution, then so are all scalar multiples $(\lambda a,\lambda
b,\lambda c) $ so we may as well assume that the Z$coordinate is equal
to 1 and are then interested in finding the solutions tex \in
\mathbb{C}^2 $ to the equation $X^3Y+Y^3+X=0 $ which gives us an affine
patch of the Kleinian quartic (in fact, these solutions give us all
points except for two, corresponding to the _points at infinity_ needed
to make the picture compact so that we can hold it in our hand and look
at it from all sides. These points at infinity correspond to the trivial
solutions (1,0,0) and (0,1,0)).

What is the connection
between points on this Riemann surface and representations? Well, if
(a,b) is a solution to the equation $X^3Y+Y^3+X=0 $, then we have a
_one-dimensional representation_ of the affine _coordinate ring_
$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) $, that is, an algebra
morphism

$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) \rightarrow \mathbb{C} $

defined by sending X to a and Y to b.
Conversely, any such one-dimensonal representation gives us a solution
(look at the images of X and Y and these will be the coordinates of
a solution). Thus, commutative algebraic geometry of smooth
curves (that is Riemann surfaces if you look at the ‘real’ picture)
can be seen as the study of one-dimensional representations of their
smooth coordinate algebras. In other words, the classical Riemann
surface gives us already the classifcation of all one-dimensional
representations, so now we are after the ‘other ones’.

In
noncommtative algebra it is not natural to restrict attention to algebra
maps to $\mathbb{C} $, at least we would also like to include algebra
maps to $n \times n $ matrices $M_n(\mathbb{C}) $. An n-dimensional
representation of the coordinate algebra of the Klein quartic is an
algebra map

$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) \rightarrow M_n(\mathbb{C}) $

That is, we want to find all pairs of $n \times n $ matrices A and B satisfying the following
matrix-identities

$A.B=B.A $ and $A^3.B+B^3+A=0_n $

The
first equation tells us that the two matrices must commute (because we
took commuting variables X and Y) and the second equation really is
a set of $n^2 $-equations in the matrix-entries of A and
B.

There is a sneaky way to get lots of such matrix-couples
from a given solution (A,B), namely by _simultaneous conjugation_.
That is, if $C \in GL_n(\mathbb{C}) $ is any invertible $n \times n $
matrix, then also the matrix-couple $~(C^{-1}.A.C,C^{-1}.B.C) $
satisfies all the required equations (write the equations out and notice
that middle terms of the form $C.C^{-1} $ cancel out and check that one
then obtains the matrix-identities

$C^{-1} A B C = C^{-1} BA C $ and $C^{-1}(A^3B+B^3+A)C = 0_n $

which are satisfied because
(A,B) was supposed to be a solution). We then say that these two
n-dimensional representations are _isomorphic_ and naturally we are
only interested in classifying the isomorphism classes of all
representations.

Using classical commutative algebra theory of
Dedekind domains (such as the coordinate ring $\mathbb{C}[X,Y]/(X^3Y+Y^3+X) $)
allows us to give a complete solution to this problem. It says that any
n-dimensional representation is determined up to isomorphism by the
following geometric/combinatorial data

  • a finite set of points $P_1,P_2,\dots,P_k $ on the Riemann surface with $k \leq n $.
  • a set of positive integers $a_1,a_2,\dots,a_k $ associated to these pointssatisfying $a_1+a_2+\dots_a_k=n $.
  • for each $a_i $ a partition of $a_i $ (that is, a decreasing sequence of numbers with total sum
    $a_i $).

To encode this classification I’ll use the mental
picture of associating to every point of the Klein quartic a small
loop. $\xymatrix{\vtx{}
\ar@(ul,ur)} $ Don\’t get over-exited about this
noncommutative manifold picture of the Klein quartic, I do not mean to
represent something like closed strings emanating from all points of the
Riemann surface or any other fanshi-wanshi interpretation. Just as
Feynman-diagrams allow the initiated to calculate probabilities of
certain interactions, the noncommutative manifold allows the
initiated to classify finite dimensional representations.

Our
mental picture of the noncommutative manifold of the Klein quartic, that
is : the points of the Klein quartic together with a loop in each point,
will tell the initiated quite a few things, such as : The fact
that there are no arrows between distict points, tells us that the
classification problem splits into local problems in a finite number of
points. Technically, this encodes the fact that there are no nontrivial
extensions between different simples in the commutative case. This will
drastically change if we enter the noncommutative world…

The fact that there is one loop in each point, tells us that
the local classification problem in that point is the same as that of
classifying nilpotent matrices upto conjugation (which, by the Jordan
normal form result, are classified by partitions) Moreover,
the fact that there is one loop in each point tells us that the local
structure of simple representations near that point (that is, the points
on the Kleinian quartic lying nearby) are classified as the simple
representations of the polynmial algebra $\mathbb{C}[x] $ (which are the
points on the complex plane, giving the picture
of the Riemann sphere in each point reflecting the local
neighborhood of the point on the Klein quartic)

In general, the
noncommutative manifold associated to a noncommutative smooth algebra
will be of a similar geometric/combinatorial nature. Typically, it will
consist of a geometric collection of points and arrows and loops between
these points. This data will then allow us to reduce the classification
problem to that of _quiver-representations_ and will allow us to give
local descriptions of our noncommutative manifolds. Next time,
I’ll give the details in the first noncommutative example : the
skew-group algebra of a finite group of automorphisms on a Riemann
surface (such as the simple group $PSL_2(\mathbb{F}_7) $ acting on the
Klein quartic). Already in this case, some new phenomena will
appear…

ADDED : While writing this post
NetNewsWire informed me that over at Noncommutative Geometry they have a
post on a similar topic : What is a noncommutative space.

One Comment

bookmarks tuesday cleanup


Geeky Mom : Why am I blogging?
. Been there before. Sooner or later
all non-pseudonomenous bloggers are faced with the same dilemmas.
There’s really no answer or advice to give except : blog when you feel
like it, if not do something different, after all its just one of those
billion of blogs around.

Texmaker : another
LaTeX-frontend, possibly having a few extras such as : a structure-pane
including labels you gave to formulas, theorems etc. (click on them
brings you to them). Intend to use it now as I’m in another rewrite of
the never-ending-book..

Microformats : “Designed for
humans first and machines second, microformats are a set of simple, open
data formats built upon existing and widely adopted standards.” May
have another look.

Quicksilver : a recurring
link. At times when I feel learning key-strokes may save me a lot of
time I have (another) go at Quicksilver. Last week, Ive reinstalled this
blog more or less post by post and used keystrokes to send a line in the
SQL-file of the database dump of NEB as a clipping to Scrivener to
MultiMarkdown it further. I used the app Service Scrubber
to define my own key-strokes. Must have another go at Quicksilver soon.
Im sure it distinguishes ‚”power mac users” from the rest of
us.


List of GTDTools
: a good list of GTD-software. I’m probably just
too chaotic for GTD to improve my workflow but somehow I cannot resist
trying some of these things out.

LifeDEV : One of those sites that tells
me I should take GTD more seriously

DoIt : One of
these GTD-tools. It is said to go well with Quicksilver, so maybe, one
day.

Think
: Here a little seemingly completely useless tool which works well (at
least for me). No, it does not make you think, but at least it helps you
while you are thinking (or doing anything a bit focussed). Install it
and enjoy! The principle is that it just blocks out all other open
windows (and there are keystrokes (yes, again) to get you quickly in
and out.) Besides, it looks great. It’s in my dock and this says it
all

Thinkature :
a brainstorming tool. Dont know why I did bookmark this. Perhaps one
day, a few years from now

Stafford Talk :
a talk by Toby Stafford I came across by accident. Maybe there are other
interesting talks on the site?

Science Scouts : a great
idea! Give yourself badges for how well you do science (or talk/write
about science). Have to collect my badges soon. I’m sure this only
works for people with a scouting-history, but who
knows?

MacResearch : Here’s a site
that may become useful. MacResearch.org is an open and independent
community for scientists using Mac OS X and related hardware in their
research. It is the mission of this site to cultivate a knowledgeable
and vibrant community of researchers to exchange ideas and information,
and collectively escalate the prominence of Apple technologies in the
scientific research community. They have some interesting articles
and tutorials on e.g. DevonThink and BibDesk etc. Worth to
revisit.

Jennifer in love : well‚ should I say something about this?
probably best not.


Breakthrough CLI
: another pamphlet in favor of the Command Line! A
must read for those who perfer GUIs to CLIs.

<

p>CLI – the
site
: Rod is working hard on CLI-20. Whenever he releases version
2.0, neverendingbooks will be among the first sites to run it. I still
love the idea.

Why do I bother? : an n-category post I got briefly interested in,
but was somehow flooded by professional
math-philosophers

Newton Legacy Reviewed : just that, a first review
on the next bookmark.

the Newton
Legacy
: a free online book, a murder mystery with a physics touch.
Perhaps this is the best investment of time/energy : write a popular
science book rather than another paper. Read half way through it (sorry
but not the best prose Ive read so far), may continue but was held up
reading a (real) murder mystery Equinox featuring also Newton and
alchemy (must be in the air somehow), also not the best mystery read
so far

Stalking with Googleearth
: no comment

(to be continued)

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