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