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_G,79 hadron physics at LHC?. I glue the comment below.

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 Γ₀(p)₊ of Γ₀(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ₀(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.

1. Congruence group Γ₀(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 Γ₀(p)₊ in SL(2,C) are identified.
   
   
2. Normalizer Γ₀(p)₊ is the subgroup of SL(2,R) commuting with Γ₀(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.
   

First some comments inspired by the observation about Gaussian Mersennes by Stephen.

1. 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).
   
   

2. The Gaussian Mersennes below M₈₉ 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= 2⁴⁶× 3²⁰× 5⁹× 7⁶× 11²× 13³× 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× 10³ TeV=17.6 PeV (P for Peta). Pessimistic would argue that this scale is the Monster energy scale never reached by human particle physicists.
   

The continuations of congruence groups and their normalizers to the p-adic variants SL(2,Z_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 H³ (mass shell for particle physicist and cosmic time constant section for cosmologist).

1. One can construct hyperbolic manifolds as spaces of the orbits of discrete subgroups in 3-D hyperbolic space H³ if the discrete subgroup defines tesselation/lattice of H³. These lattices are of special interest as the discretizations of the H³ 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.
   
   
2. 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 Γ₀(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.
   
   

3. 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 Γ₀(p) generalize by replacing p with prime of extension.
   
   

Above I have talked only about algebraic extensions of rationals. p-Adic numbers have however also finite-dimensional algebraic extensions, which are not induced by those of rational numbers.

1. 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.
   
   

2. 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 e² 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).
   
   
3. 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?
   
   

See the chapter More about TGD inspired cosmology of "Physics in Many-Sheeted Space-time"

p> For a summary of earlier postings see Links to the latest progress in TGD.