1. Introduction

  2. Two Galois groups

    1. Internal Galois group

    2. Outer Galois group

  3. Symmetries and dynamical symmetries

    1. Maps g:C2→ C2 as dynamical symmetries

    2. About the identification of the Lie groups appearing in Langlands duality?

    3. Physical interpretation of the generalized modular group and spectrum generating group

    4. Langlands duality for the representations of the Lorentz group

  4. Quantum arithmetics

    1. Functional (quantum) counterparts of integers, rational and algebraic numbers

    2. About the notion of functional primeness

    3. The notion of functional p-adics

  5. Infinite primes, the notion of rational prime, and holography= holomorphy principle

    1. The construction of infinite primes

    2. Infinite primes and holography= holomorphy principle

    3. Hierarchies of functional composites of g: C2→ C2

  6. Some questions related to the maps g

    1. What could happen in the transition f→ g\circ f?

    2. About the interpretation of the inverses of the maps g

    3. Could one understand p-adic length scale hypothesis?

    4. Critical summary of the problems associated with the physical interpretation of the number theoretical vision

  7. Infinite primes, the notion of rational prime, and holography= holomorphy principle

    1. The construction of infinite primes

    2. Infinite primes and holography= holomorphy principle

    3. Hierarchies of functional composites of g: C2→ C2

  8. Some questions related to the maps g

    1. What could happen in the transition f→ g\circ f?

    2. About the interpretation of the inverses of the maps g

    3. Could one understand p-adic length scale hypothesis?

  9. Appendix: Ramified primes for the iterates gpº n