I received a link to a popular article published in Quanta Magazine with title ‘Amazing’ Math Bridge Extended Beyond Fermat’s Last Theorem suggesting that Fermat's last theorem could generalize and provide a bridge between two very different pieces of mathematics suggested also by Langlands correspondence.
I would be happy to have the technical skills of real number theorist but I must proceed using physical analogies. What the theorem states is that one has two quite different mathematical systems, which have a deep relationship between each other.
What the discovery could be? I can make only humble guesses. The popular article tells that the "clock solutions" of given Diophantine equation in various finite fields Fp are in correspondence with the "energy" spectra of some automorphic form defined in some space.
- Diophantine equations for natural numbers, which are determined by polynomials. Their solutions can be regarded as roots of a polynomial P(x) containing second variable y as parameter. The roots which are pairs of integers are of interest now. One could consider also all roots as function of y.
- Second system consists of automorphic functions in lattice like systems, tesselations. They are encountered in Langlands conjecture, whose possible physical meaning I still fail to really understand.
The hyperboloid L ( L for Lobatchevski space) defined as t2-x2-y2-z2=constant surface of Minkowski space (particle physicist talks about mass shell) is good example about this kind of system. One can define in this kind of tesselation automorphic functions, which are quasiperiodic in sense that the values of function are fixed once one knows them for single cell of the lattice. Bloch waves serve as condensed matter analog.
One can assign to automorphic function what the article calls its "energy spectrum". In the case of hyperboloid it could correspond to the spectrum of d'Alembertian - this is physicist's natural guess. Automorphic function could be analogous to a partition function build from basic building brickes invariant under the sub-group of Lorentz group leaving the fundamental cell invariant. Zeta function assignable to extension of rationasl as generaliztion of Riemann zeta is one example.
The problem of finding the automorphic forms is difficult and the message is that here a great progress has occurred. So called torsion coefficients for the modular form would correspond the integer value roots of Diophantine equations for various finite fields Fp. What could this statement mean?
Trying to understand basic concepts
Consider first basic concepts.
One can assign to automorphic forms so called torsion coefficients in various finite fields Fp and to the eigen functions of d'Alembertian and other Casimir operators in coset space G/K. Consider discrete but infinite subgroup Γ such that solutions are apart from the factor j(γ) of automorphy left invariant under Γ. For trivial j(γ) they would be defined in double coset space Γ ∖ G/K. Besides this Galois group represented as finite discrete subgroup of SU(2) would leave the eigenfunctions invariant.
- What does automorphic form mean? One has a non-compact group G and functions from G to some vector space V. For instance, spinor modes could be considered. Automorphic forms are eigenfunctions of Casimir operators of G, d'Alembert type operator is one such operator and in TGD framework G=SO(1,3) is the interesting group to consider. There is also discrete infinite subgroup Γ⊂ G under which the eigenfunctions are not left invariant but transform by factor j(γ) of automorphy acting as matrix in V - one speaks of twisted representation.
Basic space of this kind of is upper half plane of complex plane in which G=SL(2,C) acts as also does γ=SL(2,Z) and various other discrete subgroups of SL(2,C) and defines analog of lattice consisting of fundamental domains γ∖ G as analogs of lattice cells. 3-D hyperboloid of M4 allows similar structures and is especially relevant from TGD point of view. When j(γ) is non-trivial one has analogy of Bloch waves.
Modular invariant functions is second example. They are defined in the finite-D moduli space for the conformal structures of 2-D surfaces with given genus. Automorphic forms transform by a factor j(γ) under modular transformations which do not affect the conformal equivalence class. Modular invariants formed from the modular forms can be constructed from these and the TGD based proposal for family replication phenomenon involves this kind invariants as elementary particle vacuum functions in the space of conformal equivalence classes of partonic 2-surfaces (see this).
One can also pose invariance under a compact group K acting on G from right so that one has automorphic forms
in G/K. In the case of SO(3,1) this would give automorphic forms on hyperboloid H3 ("mass shell") and this is of special interest in TGD. One could also require invariance under discrete finite subgroup acting from the left so that j(γ)=1 would be true for these transformations. Here especially interesting is the possibility that Galois group of extension of rationals is represented as this group. The correct prediction of Newton's constant from TGD indeed assumes this (see this).
- What does the spectrum (see this) mean? Spectrum would be defined by the eigenvalues of Casimir operators of G: simplest of them is analog of d'Alembertian for say SO(3,1). The number of these operators equals to the dimension of Cartan sub-algebra of G. Additional condition is posed by the transformation properties under Γ characterized by j(γ).
The analog for Diophantine equations in TGD
- Torsion group T is for the first homotopy group Π1 (fundamental group) a finite Abelian subgroup decomposing Zn to direct summands Zp, p prime. The fundamental group in the recent case would be naturally that of double coset space Γ∖ G/K.
- What could torsion coefficients be (see this)? Π1 is Abelian an representable as a product T× Zs. Zs is the dimension of Π1 - rank - as a linear space over Z and T=Zm1× Zm2×....Zmn is the torsion subgroup. The torsion coefficients mi satisfy the conditions m1⊥ m2⊥... ⊥ mn. The torsion coefficients in Fp would be naturally mi mod p.
The torsion coefficients characterize also the automorphic functions since they characterize the first homotopy group of Γ ∖ G/K . If I have understood correctly, torsion coefficients mi for various finite fields Fp for given automorphic form correspond to a sequence of solutions of Diophantine equation in Fp. This is the bridge.
- How are the Galois groups related to this (see this)? Representations of Galois group Gal(F) for finite-D extension F of rationals could act as a discrete finite subgroup of SO(3)⊂ SO(1,3) and would leave eigenfunctions invariant: these ADE groups form appear in McKay correspondence and in inclusion hierarchy of hyper-finite factors of type II1 (see this).
The invariance under Gal(F) would correspond to a special case of what I call Galois confinement, a notion that I have considered earlierhere) with physical motivations coming partially from the TGD based model of genetic code based on dark photon triplets.
The problem is to understand how dark photon triplets occur as asymptotic states - one would expect many-photon states with single photon as a basic unit. The explanation would be completely analogous to that for the appearance of 3-quark states as asymptotic states in hadron physics - the analog of color confinement. Dark photons would form Z3 triplets under Z3 subgroup of Galois group associated with corresponding space-time surface, and only Z3 singlets realized as 3-photon states would be possible.
Mathematicians talk also about the Galois group Gal(Q) of algebraic numbers regarded as an extension of finite extension F of rationals such that the Galois group Gal(F) would leave eigenfunctions invariant - this would correspond to what I have called Galois confinement.
- In TGD framework Galois group Gal(F) has natural action on the cognitive representation identified as a set of points of space-time surface for which preferred imbedding space coordinates belong to given extension of rationals (see this). In general case the action of Galois group gives a cognitive representation related to a new space-time surface, and one can construct representations of Galois group as superpositions of space-time surfaces and they are effectively wave functions in the group algebra of Gal(F). Also the action of discrete subgroup of SO(3)⊂ SO(1,3) gives a new space-time surface.
There would be two actions of Gal(F): one at the level of imbedding spaces at H3 and second at the level of cognitive representations. Possible applications of Langlands correspondence and generalization of Fermat's last theorem in TGD framework should relate to these two representations. Could the action of Galois group on cognitive representation be equivalent with its action as a discrete subgroup of SO(3)⊂ SO(1,3)? This would mean concrete geometric constraint on the preferred extremals.
What could this discovery have to do with TGD?
The analog for automorphic forms in TGD
- In adelic physics of TGD M8-H duality is in key role. Space-time surfaces can be regarded either as algebraic 4-surfaces in complexified M8 determined as roots of polynomial equations. Second representation is as mimimal surfaces with 2-D singularities identified as preferred extremals of action principle: analogs of Bohr orbits are in question.
- The Diophantine equations generalize. One considers the roots of polynomials with rational coefficients and extends them to 4-D space-time surfaces defined as roots of their continuations to octonion polynomials in the space of complexified octonions. Associativity is basic dynamical principle: the tangent space of these surfaces is quaternionic. Each irreducible polynomial defines extension of rationals via its roots and one obtains a hierarchy of them having physical interpretation as evolutionary hierarchy. These surface can be mapped to surface in H= M4×CP2 by M8-H duality.
- So called cognitive representations for given space-time surface are identified as set of points for which points have coordinate in extension of rationals. They realize the notion of finite measurement resolution and scattering ampludes can be expressed using the data provided by cognitive representations: this is extremely strong form of holography.
- Cognitive representation generalizes the solutions of Diophantine equation: instead of integers one allows points in given extension of rationals. These cognitive representations determine the information that conscious entity can have about space-time surface. As the extensions approaches algebraic numbers, the information is maximal since cognitive representation defines a dense set of space-time surface.
What is the relation to Langlands conjecture (LC)?
- The above mentioned hyperboloids of M4 are central in zero energy ontology (ZEO) of TGD: in TGD based cosmology they correspond to cosmological time constant surfaces. Also the tesselations of hyperboloids are expected to have a deep physical meaning - quantum coherence even in cosmological scales is possible and there are pieces of evidence about the lattice like structures in cosmological scales.
- Also the finite lattices defined by finite discrete subgroups of SU(3) in CP2 analogous to Platonic solids and and regular polygons for rotation group are expected to be important.
- One can imagine analogs of automorphic forms for these tesselations. The spectrum would correspond to that for massless d'Alembertian of L×CP2, where L denotes the hyperboloid, satisfying the boundary conditions given by tesselation. In condensed matter physics solutions of Schroedinger equation consistant with lattice symmetries would be in question: Bloch waves. The spectrum would correspond to mass squared eigenvalues and to the spectra for observables assignable to the discrete subgroup of Lorentz group defining the tesselation.
- The theorem described in the article suggests a generalization in TGD framework based on physical motivations. The "energy" spectrum of these automorphic forms identified as mass squared eigenvalues and other quantum numbers characterized by the subgroup of Lorentz group are at the other side of the bridge.
At the other side of bridge could be the spectrum of the roots of polynomials defining space-time surfaces. A more general conjecture would be that the discrete cognitive representations for space-time surfaces as "roots" of octonionic polynomial are at the other side of bridge. These two would correspond to each other.
Cognitive representations at space-time level would code for the spectrum of d'Alembertian like operator at the level of imbedding space. This could be seen as example of quantum classical correspondence (QCC) , which is basic principle of TGD.
I understand very little about LC at technical level but I can try to relate it to TGD via physical analogies.
To sum up, Langlands correspondence would relate two descriptions. Discrete description for cognitive representations at space-time level and continuum description at imbedding space level in terms of eigenfunctions of spinor d'Alembertian.
- LC relates two kinds of groups.
- Algebraic groups satisfying certain very general additional conditions (complex nxn matrices is one example). Matrix groups such as Lorentz group are a good example.
The Cartesian product of future light-cone and CP2 would be the basic space. d'Alembertian inside future light-cone in the variables defined by Robertson- Walker coordinates. The separation of variables a as light-cone proper time and coordinates of H3 for given value of a assuming eigenfunction of H3 d'Alembertian satisfying additional symmetry conditions would be in question. The dependence on a is fixed by the separability and by the eigenvalue value of CP2 spinor Laplacian.
- So called L-groups assigned with extensions of rationals and function fields defined by algebraic surfaces as as those defined by roots of polynomials. This brings in adelic physics in TGD.
- The physical meaning in TGD could be that the discrete the representations provided by the extensions of rationals and function fields on algebraic surfaces (space-time surfaces in TGD) determined by them. Function fields might be assigned to the modes of induce spinor fields.
The physics at the level of imbedding space (M8 or H) described in terms of real and complex numbers - the physics as we usually understand it - would by LC corresponds to the physics provided by discretizations of space-time surfaces as algebraic surfaces. This correspondence would not be 1-1 but many-to-one. Discretization provided by cognitive representations would provide hierarchy of approximations. Langlands conjecture would justify this vision.
- Galois groups of extensions are excellent examples of L-groups an indeed play central role in TGD. The proposal is that Galois groups provide a representation for the isometries of the imbedding space and also for the hierarchy of dynamically generated symmetries. This is just what the Langlands conjecture motivates to say.
Amusingly, just last week I wrote an article deducing the value of Newton's constant using the conjecture that discrete subgroup of isometries common to M8 and M4×CP2 consisting of a product of icosahedral group with 3 copies of its covering corresponds to Galois group for extension of rationals. The prediction is correct (see this). The possible connection with Langlands conjecture came into my mind while writing this.
See the chapter Langlands Program and TGD: Years Later or the article Generalization of Fermat's last theorem and TGD.