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Tag: Ono

Borcherds’ favourite numbers

Whenever I visit someone’s YouTube or Twitter profile page, I hope to see an interesting banner image. Here’s the one from Richard Borcherds’ YouTube Channel.

Not too surprisingly for Borcherds, almost all of these numbers are related to the monster group or its moonshine.

Let’s try to decode them, in no particular order.

196884

John McKay’s observation 196884=1+196883 was the start of the whole ‘monstrous moonshine’ industry. Here, 1 and 196883 are the dimensions of the two smallest irreducible representations of the monster simple group, and 196884 is the first non-trivial coefficient in Klein’s j-function in number theory.

196884 is also the dimension of the space in which Robert Griess constructed the Monster, following Simon Norton’s lead that there should be an algebra structure on the monster-representation of that dimension. This algebra is now known as the Griess algebra.

Here’s a recent talk by Griess “My life and times with the sporadic simple groups” in which he tells about his construction of the monster (relevant part starting at 1:15:53 into the movie).

1729

1729 is the second (and most famous) taxicab number. A long time ago I did write a post about the classic Ramanujan-Hardy story the taxicab curve (note to self: try to tidy up the layout of some old posts!).

Recently, connections between Ramanujan’s observation and K3-surfaces were discovered. Emory University has an enticing press release about this: Mathematicians find ‘magic key’ to drive Ramanujan’s taxi-cab number. The paper itself is here.

“We’ve found that Ramanujan actually discovered a K3 surface more than 30 years before others started studying K3 surfaces and they were even named. It turns out that Ramanujan’s work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics.”

Ken Ono

24

There’s no other number like 24 responsible for the existence of sporadic simple groups.

24 is the length of the binary Golay code, with isomorphism group the sporadic Mathieu group M24 and hence all of the other Mathieu-groups as subgroups.

24 is the dimension of the Leech lattice, with isomorphism group the Conway group Co0=.0 (dotto), giving us modulo its center the sporadic group Co1=.1 and the other Conway groups Co2=.2,Co3=.3, and all other sporadics of the second generation in the happy family as subquotients (McL,HS,Suz and HJ=J2)



24 is the central charge of the Monster vertex algebra constructed by Frenkel, Lepowski and Meurman. Most experts believe that the Monster’s reason of existence is that it is the symmetry group of this vertex algebra. John Conway was one among few others hoping for a nicer explanation, as he said in this interview with Alex Ryba.

24 is also an important number in monstrous moonshine, see for example the post the defining property of 24. There’s a lot more to say on this, but I’ll save it for another day.

60

60 is, of course, the order of the smallest non-Abelian simple group, A5, the rotation symmetry group of the icosahedron. A5 is the symmetry group of choice for most viruses but not the Corona-virus.

3264

3264 is the correct solution to Steiner’s conic problem asking for the number of conics in PC2 tangent to five given conics in general position.



Steiner himself claimed that there were 7776=65 such conics, but realised later that he was wrong. The correct number was first given by Ernest de Jonquières in 1859, but a rigorous proof had to await the advent of modern intersection theory.

Eisenbud and Harris wrote a book on intersection theory in algebraic geometry, freely available online: 3264 and all that.

248

248 is the dimension of the exceptional simple Lie group E8. E8 is also connected to the monster group.

If you take two Fischer involutions in the monster (elements of conjugacy class 2A) and multiply them, the resulting element surprisingly belongs to one of just 9 conjugacy classes:

1A,2A,2B,3A,3C,4A,4B,5A or 6A

The orders of these elements are exactly the dimensions of the fundamental root for the extended E8 Dynkin diagram.

This is yet another moonshine observation by John McKay and I wrote a couple of posts about it and about Duncan’s solution: the monster graph and McKay’s observation, and E8 from moonshine groups.

163

163 is a remarkable number because of the ‘modular miracle’
eπ163=262537412640768743.99999999999925
This is somewhat related to moonshine, or at least to Klein’s j-function, which by a result of Kronecker’s detects the classnumber of imaginary quadratic fields Q(D) and produces integers if the classnumber is one (as is the case for Q(163)).

The details are in the post the miracle of 163, or in the paper by John Stillwell, Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.

Richard Borcherds, the math-vlogger, has an entertaining video about this story: MegaFavNumbers 262537412680768000

His description of the j-function (at 4:13 in the movie) is simply hilarious!

Borcherds connects 163 to the monster moonshine via the j-function, but there’s another one.

The monster group has 194 conjugacy classes and monstrous moonshine assigns a ‘moonshine function’ to each conjugacy class (the j-function is assigned to the identity element). However, these 194 functions are not linearly independent and the space spanned by them has dimension exactly 163.

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Monstrous dessins 1

Dedekind’s Psi-function Ψ(n)=np|n(1+1p) pops up in a number of topics:

  • Ψ(n) is the index of the congruence subgroup Γ0(n) in the modular group Γ=PSL2(Z),
  • Ψ(n) is the number of points in the projective line P1(Z/nZ),
  • Ψ(n) is the number of classes of 2-dimensional lattices LMgh at hyperdistance n in Conway’s big picture from the standard lattice L1,
  • Ψ(n) is the number of admissible maximal commuting sets of operators in the Pauli group of a single qudit.

The first and third interpretation have obvious connections with Monstrous Moonshine.

Conway’s big picture originated from the desire to better understand the Moonshine groups, and Ogg’s Jack Daniels problem
asks for a conceptual interpretation of the fact that the prime numbers such that Γ0(p)+ is a genus zero group are exactly the prime divisors of the order of the Monster simple group.

Here’s a nice talk by Ken Ono : Can’t you just feel the Moonshine?



For this reason it might be worthwhile to make the connection between these two concepts and the number of points of P1(Z/nZ) as explicit as possible.

Surely all of this is classical, but it is nicely summarised in the paper by Tatitscheff, He and McKay “Cusps, congruence groups and monstrous dessins”.

The ‘monstrous dessins’ from their title refers to the fact that the lattices LMgh at hyperdistance n from L1 are permuted by the action of the modular groups and so determine a Grothendieck’s dessin d’enfant. In this paper they describe the dessins corresponding to the 15 genus zero congruence subgroups Γ0(n), that is when n=1,2,3,4,5,6,7,8,9,10,12,13,16,18 or 25.

Here’s the ‘monstrous dessin’ for Γ0(6)



But, one can compute these dessins for arbitrary n, describing the ripples in Conway’s big picture, and try to figure out whether they are consistent with the Riemann hypothesis.

We will get there eventually, but let’s start at an easy pace and try to describe the points of the projective line P1(Z/nZ).

Over a field k the points of P1(k) correspond to the lines through the origin in the affine plane A2(k) and they can represented by projective coordinates [a:b] which are equivalence classes of couples (a,b)k2{(0,0)} under scalar multiplication with non-zero elements in k, so with points [a:1] for all ak together with the point at infinity [1:0]. When n=p is a prime number we have #P1(Z/pZ)=p+1. Here are the 8 lines through the origin in A2(Z/7Z)



Over an arbitrary (commutative) ring R the points of P1(R) again represent equivalence classes, this time of pairs
(a,b)R2 : aR+bR=R
with respect to scalar multiplication by units in R, that is
(a,b)(c,d)  iff λR : a=λc,b=λd
For P1(Z/nZ) we have to find all pairs of integers (a,b)Z2 with 0a,b<n with gcd(a,b)=1 and use Cremona’s trick to test for equivalence:
(a,b)=(c,d)P1(Z/nZ) iff adbc0 mod n
The problem is to find a canonical representative in each class in an efficient way because this is used a huge number of times in working with modular symbols.

Perhaps the best algorithm, for large n, is sketched in pages 145-146 of Bill Stein’s Modular forms: a computational approach.

For small n the algorithm in §1.3 in the Tatitscheff, He and McKay paper suffices:

  • Consider the action of (Z/nZ) on {0,1,,n1}=Z/nZ and let D be the set of the smallest elements in each orbit,
  • For each dD compute the stabilizer subgroup Gd for this action and let Cd be the set of smallest elements in each Gd-orbit on the set of all elements in Z/nZ coprime with d,
  • Then P1(Z/nZ)={[c:d] | dD,cCd}.

Let’s work this out for n=12 which will be our running example (the smallest non-squarefree non-primepower):

  • (Z/12Z)={1,5,7,11}C2×C2,
  • The orbits on {0,1,,11} are
    {0},{1,5,7,11},{2,10},{3,9},{4,8},{6}
    and D={0,1,2,3,4,6},
  • G0=C2×C2, G1={1}, G2={1,7}, G3={1,5}, G4={1,7} and G6=C2×C2,
  • 1 is the only number coprime with 0, giving us [1:0],
  • {0,1,,11} are all coprime with 1, and we have trivial stabilizer, giving us the points [0:1],[1:1],,[11:1],
  • {1,3,5,7,9,11} are coprime with 2 and under the action of {1,7} they split into the orbits
    {1,7}, {3,9}, {5,11}
    giving us the points [1:2],[3:2] and [5:2],
  • {1,2,4,5,7,8,10,11} are coprime with 3, the action of {1,5} gives us the orbits
    {1,5}, {2,10}, {4,8}, {7,11}
    and additional points [1:3],[2:3],[4:3] and [7:3],
  • {1,3,5,7,9,11} are coprime with 4 and under the action of {1,7} we get orbits
    {1,7}, {3,9}, {5,11}
    and points [1:4],[3:4] and [5,4],
  • Finally, {1,5,7,11} are the only coprimes with 6 and they form a single orbit under C2×C2 giving us just one additional point [1:6].

This gives us all 24=Ψ(12) points of P1(Z/12Z) (strangely, op page 43 of the T-H-M paper they use different representants).

One way to see that #P1(Z/nZ)=Ψ(n) comes from a consequence of the Chinese Remainder Theorem that for the prime factorization n=p1e1pkek we have
P1(Z/nZ)=P1(Z/p1e1Z)××P1(Z/pkekZ)
and for a prime power pk we have canonical representants for P1(Z/pkZ)
[a:1] for a=0,1,,pk1 and[1:b] for b=0,p,2p,3p,,pkp
which shows that #P1(Z/pkZ)=(p+1)pk1=Ψ(pk).

Next time, we’ll connect P1(Z/nZ) to Conway’s big picture and the congruence subgroup Γ0(n).

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a non-commutative Jack Daniels problem

At a seminar at the College de France in 1975, Tits wrote down the order of the monster group

#M=246.320.59.76.112.133.17·19·23·29·31·41·47·59·71

Andrew Ogg, who attended the talk, noticed that the prime divisors are precisely the primes p for which the characteristic p super-singular j-invariants are all defined over Fp.

Here’s Ogg’s paper on this: Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 16, no 1 (1974-1975).

Ogg offered a bottle of Jack Daniels for an explanation of this coincidence.

Even Richard Borcherds didn’t claim the bottle of Jack Daniels, though his proof of the monstrous moonshine conjecture is believed to be the best explanation, at present.

A few years ago, John Duncan and Ken Ono posted a paper “The Jack Daniels Problem”, in which they prove that monstrous moonshine implies that if p is not one of Ogg’s primes it cannot be a divisor of #M. However, the other implication remains mysterious.

Duncan and Ono say:

“This discussion does not prove that every pOgg divides #M. It merely explains how the first principles of moonshine suggest this implication. Monstrous moonshine is the proof. Does this then provide a completely satisfactory solution to Ogg’s problem? Maybe or maybe not. Perhaps someone will one day furnish a map from the characteristic p supersingular j-invariants to elements of order p where the group structure of M is apparent.”

I don’t know whether they claimed the bottle, anyway.

But then, what is the non-commutative Jack Daniels Problem?

A footnote on the first page of Conway and Norton’s ‘Monstrous Moonshine’ paper says:

“Very recently, A. Pizer has shown these primes are the only ones that satisfy a certain conjecture of Hecke from 1936 relating modular forms of weight 2 to quaternion algebra theta-series.”

Pizer’s paper is “A note on a conjecture of Hecke”.

Maybe there’s a connection between monstrous moonshine and the arithmetic of integral quaternion algebras. Some hints:

The commutation relations in the Big Picture are reminiscent of the meta-commutation relations for Hurwitz quaternions, originally due to Conway in his booklet on Quaternions and Octonions.

The fact that the p-tree in the Big Picture has valency p+1 comes from the fact that the Brauer-Severi of M2(Fp) is PFp1. In fact, the Big Picture should be related to the Brauer-Severi scheme of M2(Z).

Then, there’s Jorge Plazas claiming that Connes-Marcolli’s GL2-system might be related to moonshine.

One of the first things I’ll do when I return is to run to the library and get our copy of Shimura’s ‘Introduction to the arithmetic theory of automorphic functions’.

Btw. the bottle in the title image is not a Jack Daniels but the remains of a bottle of Ricard, because I’m still in the French mountains.

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