Mathemematical bridge connecting Diophantine equations and spectrum of automorphic functions

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.

1. 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.

2. 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.

Could this have something to do with TGD?

The analog for Diophantine equations in TGD

1. 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.

2. 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.
3. 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.
4. 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.

The analog for automorphic forms in TGD

1. 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.

2. 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.
3. 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.
4. The theorem described in the article suggests a generalization in TGD framework based on physical motivations. The “energy” spectrum of these automorphic forms idenified 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.

What is the relation to Langlands conjecture (LC)?

I understand very little about LC at technical level but I can try to relate it to TGD via physical analogies.

1. LC relates two kinds of groups.
   1. Algebraic groups satisfying certain very general additional conditions (complex nxn matrices is one example). Matrix groups such as Lorentz group are a good example.

   2. 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.
2. 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.

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. The possible connection with Langlands conjecture came into my mind just now.

See for a blog post about gravitational constant. I hope that I get pdf file to my homepage soon. There are some technical difficulties.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.