Witt vectors and Witt polynomials and the representation of p-adic numbers as space-time surfaces

The Quanta Magazine article (see this) related to Langlands correspondence and involving concepts like elliptic curves, modular functions, and Galois groups served as an inspiration for these comments. Andrew Wiles in his proof of Fermat's Last Theorem used a relationship between elliptic curves and modular forms. Wiles proved that certain kinds of elliptic curves are modular in the sense that they correspond to a unique modular form. Later it was proved that this is true for all elliptic surfaces. Later the result was generalized to real quadratic extensions of rationals by 3 mathematicians involving Samir Siksek and now by Caraiani and Newton for the imaginary quadratic extensions.

Could this correspondence be proved for all algebraic extensions of rationals? And what about higher order polynomials of two variables? Complex elliptic curves, defined as roots of third order polynomials of two complex variables, are defined in 2-D space with two complex dimensions have the special feature that they allow a 2-D discrete translations as symmetries: in other words, they are periodic for a suitable chosen complex coordinate. I have talked about this from TGD point of view in (see this). Is the 1-1 correspondence with modular forms possible only for elliptic curves having these symmetries?

How are the Galois groups related to this? Indian mathematical genius Ramanujan realized that modular forms seem to be associated with so-called Galois representations. The Galois group would be the so- called absolute Galois group of the number field involved with the representation. Very roughly, they could be seen as representations of a Lie group which extends the Galois group. Also elliptic curves are associated with Galois representations. This suggests that the Galois representations connect elliptic curves, objects of algebraic geometry and modular forms, which correspond to group representations. These observations led to Langlands program which roughly states a correspondence between geometry and number theory.

The Galois group is indeed involved with Langlands duality. If the Lie group G is defined over field k (in the recent case extension of rationals), the Langlands dual ^LG of G is an extension of the absolute Galois group of k by a complex Lie group (see this). The representation of the absolute Galois group is finite-dimensional, which suggests that it reduces to a Galois group for a finite-dimensional extension of rationals. Therefore the effective Galois group used can be larger than the Galois group of extension of rationals. ^LG has the same Lie algebra as G.

In the following, I will consider the situation from a highly speculative view point provided by TGD. In TGD, geometric and number theoretic visions of physics are complementary: M⁸-H duality in which M⁸ is analogous to 8-D momentum space associated with 8-D H=M⁴× CP₂ is a formulation for this duality and makes Galois groups and their generalizations dynamic symmetries in the TGD framework (see this). This complementarity is analogous to momentum position duality of quantum theory and implied by the replacement of a point-like particle with 3-surface, whose Bohr orbit defines space-time surface.

At a very abstract level this view is analogous to Langlands correspondence (see this). The recent view of TGD involving an exact algebraic solution of field equations based on holography= holomorphy vision allows to formulate the analog Langlands correspondence in 4-D context rather precisely. This requires a generalization of the notion of Galois group from 2-D situation to 4-D situation: there are 2 generalizations and both are required.

1. The first generalization realizes Galois group elements, not as automorphisms of a number field, but as analytic flows in H=M⁴× CP₂ permuting different regions of the space-time surface identified as roots for a pair f=(f₁,f₂) of pairs f=(f₁,f₂): H→ C², i=1,2. The functions f_i are analytic functions of one hypercomplex and 3 complex coordinates of H.
2. Second realization is for the spectrum generating algebra defined by the functional compositions g(f), where g: C²→ C² is analytic function of 2 complex variables. The interpretation is as a cognitive hierarchy of function of functions of .... and the pairs (f₁,f₂) which do not allow a composition of form f=g(h) correspond to elementary function and to the lowest levels of this hierarchy, kind of elementary particles of cognition. Also the pairs g can be expressed as composites of elementary functions.

   If g₁ and g₂ are polynomials with coefficients in field E identified as an extension of rationals, one can assign to g (f) root a set of pairs (r₁,r₂) as roots f₁,f₂)= (r₁,r₂) and r_i are algebraic numbers defining disjoint space-time surfaces. One can assign to the set of root pairs the analog of the Galois group as automorphisms of the algebraic extension of the field E appearing as the coefficient field of (f₁,f₂) and (g₁,g₂). This hierarchy leads to the idea that physics could be seen as analog of formal system appearing in Gödel's theorems and that the hierarchy of functional composites could correspond to a hierarchy of meta levels in mathematical cognition (see this).