More about Langlands correspondence in the TGD framework

The interview of Edward Frenkel relating to the Langlands correspondence was very inspiring and led to a considerably more detailed understanding of how number theoretic and geometric Langlands correspondence emerge in the TGD framework from number theoretic universality, holography = holomorphy vision leading to a general solution of field equations based on the generalization of holomorphy, and M⁸-H duality relating geometric and number theoretic visions of TGD.

The space-time surfaces are realized as roots for a pair (P₁,P₂) of holomorphic polynomials of four generalized complex coordinates of H=M⁴× CP₂. In this view space-time surfaces are representations of the function field of generalized polynomial pairs in H and can be regarded as numbers with arithmetic operations induced from those for the polynomial pairs.

A proposal for how to count the number of roots of the (P₁,P₂)=(0,0), when the arguments are restricted to a finite field in terms of modular forms defined at the hyperboloid H³× CP₂ ⊂ M⁴× CP₂. The geometric variant of the Galois group as a group mapping different roots for a polynomial pair (P₁,P₂) identifiable as regions of the space-time surface (minimal surface) would be in terms of generalized holomorphisms of H.

See the article About Langlands correspondence in the TGD framework or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.