### Congruence subgroups of SL(2,R), Monster Moonshine, Gaussian Mersennes, and p-adic physics

Stephen Crowley made a very interesting observation about Gaussian Mersennes in the comment section of the posting

*Pion of M*. I glue the comment below.

_{G,79}hadron physics at LHC?* Matti, why Low Gaussian primes? Your list of primes is a subset of the factors of the dimension of the friendly giant group.
*

*The monster group was investigated in the 1970s by mathematicians Jean-Pierre Serre, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ _{0}(p)_{+} of Γ_{0}(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ_{0}(p)_{+} has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg heard about the monster group later on, and noticed that these were precisely the prime factors of the size of Monster, he published a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact (Ogg (1974)).*

I must first try to clarify to myself some definitions so that I have some idea about what I am talking about.

- Congruence group Γ
_{0}(p) is the kernel of the modulo homomorphism mapping SL(2,Z) to SL(2,Z/pZ) and thus consists of SL(2,Z) matrices which are are unit matrices modulo p. More general congruence subgroups SL(2,Z/nZ) are subgroups of SL(2,Z/pZ) for primes p dividing n. Congruence group can be regarded as subgroup of p-adic variant of SL(2,Z) with elements restricted to be finite as real integers. One can give up the finiteness in real sense by introducing p-adic topology so that one has SL(2,Z_{p}). The points of hyperbolic plane at the orbits of the normalizer of Γ_{0}(p)_{+}in SL(2,C) are identified.

- Normalizer Γ
_{0}(p)_{+}is the subgroup of SL(2,R) commuting with Γ_{0}(p) but not with its individual elements. The quotient of hyperbolic space with the normalizer is sphere for primes k associated with Gaussian Mersennes up to k=47. The normalizer in SL(2,Z_{p}) would also make sense and an interesting question is whether the result can be translated to p-adic context. Also the possible generalization to SL(2,C) is interesting.

- Gaussian primes are really big but the primes defining them are logarithmically smaller. k=379 defines scale slightly large than that defined by the age of the Universe. Larger ones exist but are not terribly interesting for human physicists for a long time.

Some primes k define Gaussian Mersenne as M

_{G,k}= (1+i)^{k}-1 and the associated real prime defined by its norm is rather large - rather near to 2^{k}and for k= 79 this is already quite big. k=113 characterises muon and nuclear physics, k=151,157,163,167 define a number theoretical miracle in the range cell membrane thickness- size of cell nucleus. Besides this there are astro-physically and cosmoplogically important Gaussian Mersennes (see the earlier posting).

- The Gaussian Mersennes below M
_{89}correspond to k=2, 3, 5, 7, 11, 19, 29, 47, 73. Apart from k=73 this list is indeed contained by the list of the lowest monster primes k= 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71. The order d of Monster is product of powers of these primes: d= 2^{46}× 3^{20}× 5^{9}× 7^{6}× 11^{2}× 13^{3}× 17× 19× 23× 29× 31× 41× 47× 59× 71 .

Amusingly, Monster contains subgroup with order, which is product of exactly those primes k associated with Gaussian Mersennes, which are definitely outside the reach of LHC! Should one call this subgroup Particle Physics Monster? Number theory and particle physics would meet each other! Or actually they would not!

Speaking seriously, could this mean that the high energy physics above M

_{G,79}energy is somehow different from that below in TGD Universe? Is k=47 somehow special: it correspond to energy scale 17.6× 10^{3}TeV=17.6 PeV (P for Peta). Pessimistic would argue that this scale is the Monster energy scale never reached by human particle physicists.

_{p}) of SL(2,Z+iZ) (SL(2,C) are very interesting in TGD framework and are expected to appear in the adelization. Now hyperbolic plane is replaced with 3-D hyperbolic space H

^{3}(mass shell for particle physicist and cosmic time constant section for cosmologist).

- One can construct hyperbolic manifolds as spaces of the orbits of discrete subgroups in 3-D hyperbolic space H
^{3}if the discrete subgroup defines tesselation/lattice of H^{3}. These lattices are of special interest as the discretizations of the H^{3}parametrizing the position for the second tip of causal diamond (CD) in zero energy ontology (ZEO), when the second tip is fixed. By number theoretic arguments this moduli space should be indeed discrete.

- In TGD inspired cosmology the positions of dark astrophysical objects could tend to be localized in hyperbolic lattice and visible matter could condense around dark matter. There are infinite number of different lattices assignable to the discrete subgroups of SL(2,C). Congruence subgroups and/or their normalizers might define p-adically natural tesselations. In ZEO this kind of lattices could be also associated with the light-like boundaries of CDs obtained as the limit of hyperbolic space defined by cosmic time constant hyperboloid as cosmic time approaches zero (moment of big bang). In biology there is evidence for coordinate grid like structures and I have proposed that they might consist of magnetic flux tubes carrying dark matter.

Only a finite portion of the light-cone boundary would be included and modulo p arithmetics refined by using congruence subgroups Γ

_{0}(p) and their normalizers with the size scale of CD identified as secondary p-adic time scale could allow to describe this limitation mathematically. Γ(n) would correspond to a situation in which the CD has size scale given by n instead of prime: in this case, one would have multi-p p-padicity.

- In TGD framework one introduces entire hierarchy of algebraic extensions of rationals. Preferred p-adic primes correspond to so called ramified primes of the extension, and also p-adic length scale hypothesis can be understood and generalized if one accepts Negentropy Maximization Principle (NMP) and the notion of negentropic entanglement. Given extension of rationals induces an extension of p-adic numbers for each p, and one obtains extension of of ordinary adeles. Algebraic extension of rationals leads also an extension of SL(2,Z). Z can be replaced with any extension of rationals and has p-adic counterparts associated with p-adic integers of extensions of p-adic numbers. The notion of primeness generalizes and the congruence subgroups Γ
_{0}(p) generalize by replacing p with prime of extension.

- The basic observation is that e
^{p}exists as power series p-adically as p-adic integer of norm 1 - e^{p}cannot be regarded as a rational number. One can introduce also roots of e^{p}and define in these manner algebraic extensions of p-adic numbers. For rational numbers the extension would be algebraically infinite-dimensional.

In real number based Lie group theory e is in special role more or less by convention. In p-adic context the situation changes. p-adic variant of a given Lie group is obtained by exponentiation of elements of Lie algebra which are proportional to p (one obtains hierarchy of sub-Lie groups in powers of p) so that the Taylor series converges p-adically.

These subgroups and algebraic groups generate more interesting p-adic variants of Lie groups: they would decompose into unions labelled by the elements of algebraic groups, which are multiplied by the p-adic variant of Lie group. The roots of e are mathematically extremely natural serving as hyperbolic counterparts for the roots of unity assignable to ordinary angles necessary if one wants to talk about the notion of angle and perform Fourier analysis in p-adic context: actually one can speak only about trigonometric functions of angles p-adically but not about angles. Same is true in hyperbolic sector.

- The extension of p-adics containg roots of e could even have application to cosmology! If the dark astrophysical objects tend to form hyperbolic lattices and visible matter tends to condensed around lattice points, cosmic redshifts tend to have quantized values. This tendency is observed. Also roots of e
^{p}could appear. The recently observed evidence for the oscillation of the cosmic scale parameter could be understood if one assumes this kind of dark matter lattice, which can oscillate. Roots of e^{2}appear in the model! (see the posting*Does the rate of cosmic expansion oscillate?*). Analogous explanation in terms of dark matter oscillations applies to the recently observed anomalous periodic variations of Newton's constant measured at the surface of Earth and of the length of day (*Variation of Newton's constant and of length of day*).

- Things can get even more complex! e
^{Π}converges Π-adically for any generalized p-adic number field defined by a prime Π of an algebraic extension and one can introduce genuinely p-adic algebraic extensions by introducing roots e^{Π/n}! This raises interesting questions. How many real transcendentals can be represented in this manner? How well the hierarchy of adeles associated with extensions of rationals allowing also genuinely p-adic finite-dimensionals extensions of p-adics is able to approximate real number system? For instance, can one represent Π in this manner?

## 3 Comments:

Matti, very good post with many deeper issues in the mathematics that can lead to variations on our emphases for speculative models. He seems to be suggesting your emphasis in particular may fit into the idea that primes are subsets of the monsters. But like string theory it is not clear in the details that we can say just what measures can be made. I thought your idea was a better generalization extending the idea of particle high numbers to low ones as well. So without a deeper understanding of the dark stuff (and the transcendental's as he mentions) we can only intuit vague questions of things like cosmic expansion oscillation or Newton's constant varying. There is also Surreal Calculus principles (Conway). Pi as a ten fold symmetry base looks rather stringy with structural gaps such as the primes show rather than issues of binary dualities. Consciousness adds to the complexity of the problem. What does it take to convey such ideas? We are at the beginning of a whole new physics and evidently mathematics. I see it as a very small part of what is to come. But it was nice to see articles with thoughts like our own that may find a wider truth of things in such detail narrow as it was as a looking back at our great theoreticians modest achievements. A good summary of our ongoing bottleneck to wider theories of everything.

The point concerning monster was that the number N of its elements have primes associated with the definition of Gaussian Mersennes (1+i)^k-1, up to k=47 as prime factors. Why this is the case is mystery.

What makes this so funny is that in these primes happen to correspond to possible copies of hadron physics definitely outside the reach of LHC;-). k=73 might have been already discovered: there is 3.5 sigma evidence for 2.2 TeV bump: something completely unexpected, it could be M_{G,73} pion. Even more amusing, I realised the possibility of k=73 hadron physics besides k=89 only recently.

Deeper understanding of dark matter is the key problem of physics now. I am convinced that it requires going beyond the reductionistic view and number theoretical vision/quantum criticality having as its satellite concepts fractality, h_eff hierarchy, p-adic length scale hierarchy, adelic physics, ZEO, etc… is to my opinion sufficiently general approach.

This is deeply conceptual problem, not about hypothesising some new exotic particle. It will not be solved by master symbol manipulator but real thinker.

Varying Newton's constants, oscillating acceleration of cosmic expansion, as also recent progress in condensed matter physics suggesting that condensed matter physicists are already "seeing" many sheeted physics without knowing that this is the case, are extremely welcome empirical inputs helping to bind this picture with empirical reality.

This is like solving a huge modern crossword with some overall hidden theme and full of jokes: once you discover it, everything becomes suddenly very easy.

Matti,

I agree that this view can be a quantitative or calculable measure. The idea of Branes as such, as embedded or as multiple sheets is part of a wider interpretation ( which I think Lubos for example is coming around to a more general view which does resemble string theory where it soberly does not claim to be the total picture.) In the simulations and disconnections of implied or deeper space structures of the say vacuum or so called dark issues I do appreciate your qualitative intuitions including what happens at the brain stem... my concern is at the physics of higher brain regions where we seem born with a sense of space yet our personal orientation comes from something within. I would say off hand that what is happening here and in connection to the brain stem, and the right and left brain parity issues supports the standard theory in the string or loopy landscape... but it is in a more general level than say Lie groups which is to say that it is a topological dynamic view thus even in the superimposition of branes by group theory and issues of scale we have to reach a little higher than interpretations of Gaussian Prime ideas or any simply limited complex space structures (such particles may have been seen already as you said.) So it seems we must know the more general theory to show explicit calculations or explain why we can imagine some things or sense some things by which we do understand diverse representations of what is consciousness. I think a lot of this could be calculated more as arithmetic rather than abstract algebra. Keep up the good thinking.

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