A second attempt to understand the notions of perfectoid and its tilt

Quanta Magazine tells that 30-year old Peter Scholze is now the youngest Fields medalist due to the revolution that he launched in arithmetic geometry (see this).

Scholze's work might be interesting also from the point of view physics, at least the physics according to TGD. I have already made a attempt to understand Scholze's basic idea and to relate it to physics. About the theorems that he has proved I cannot say anything with my miserable math skills.

The notion of perfectoid

Scholze introduces first the notion of perfectoid.

1. This requires some background notions. The characteristic p for field is defined as the integer p for which px=0 for all elements x. Frobenius homomorphism (Frob familiarly) is defined as x→ x^p. For a field of characteristic p Frob: x-->x^p is an algebra homomorphism mapping product to product and sum to sum: this is very nice and relatively easy to show even by a layman like me.
2. Perfectoid is a field having either characteristic p=0 (reals, p-adics for instance) or for which Frob is a surjection meaning that Frob maps at least one number to a given number x.

   For finite fields Frob is identity: x^p=x as proved already by Fermat. For reals and p-adic number fields with characteristic p=0 Frob maps all elements to unit element and is not a surjection but this is not required now.

The tilt of the perfectoid

What Scholze introduces besides perfectoids K also what he calls tilt of the perfectoid: K_b. K_b is something between p-adic number fields and reals and leads to theorems giving totally new insights to arithemetic geometry

1. As we learned during the first student year, real numbers can be defined as Cauchy sequences of rationals converging to a real number, which can be also algebraic number or transcendental. The numbers in the tilt K_b would be this kind of sequences.
2. Scholze starts from (say) p-adic numbers and considers infinite sequence of iterates of 1/p:th roots. At given step x→ x^1/p. This gives the sequences $(x,x^1/p,x^1/p2,x^1/p3,...) as elements of K_b. At the limit one obtains 1/p^∞ root of x.
   1. For finite fields each step is trivial (x^p=x) so that nothing interesting results: one has (x,x,x,x,x...).
   2. For p-adic number fields the situation is non-trivial. x^1/p exists as p-adic number for all p-adic numbers with unit norm having x= x₀+x₁p+... In the lowest order x ≈ x₀ the root is just x since x is effectively an element of finite field in this approximation. One can develop the x^1/pto a power series in p and continue the iteration. The sequence obtained defines the element of tilt K_b of field K, now p-adic numbers.
   3. If the p-adic number x has norm pⁿ and is therefore not unity, the root operation makes sense only if one performs an extension of p-adic numbers containing all the roots p^1/pk) . These roots define one particular kind of extension of p-adic numbers and the extension is infinite-dimensional since all roots are needed. One can approximate K_b by taking only finite number iterated roots: I call these almost perfectoids as precursors of perfectoids.
3. The tilt is said to be fractal: this is easy to understand from the presence of the iterated p:th root. Each step in the sequence is like zooming. One might say that p-adic scale p becomes p:th root of itself. In TGD p-adic length scale L_p is proportional to p^1/2: does the scaling mean that the p-adic length scale would defined a hierarchy of scales proportional to p^1/2kp approaching the CP₂ scale since the root of p approaches unity. Tilts as extensions by iterated roots would improve the length scale resolution.

Characteristic p (p is now the prime labelling p-adic number field) means nx=0. This property makes the mathematics of finite fields extremely simple: in the summation one need not take care of the residue as in the case of reals and p-adics. The tilt of the p-adic number field would have the same property! In the infinite sequence of the p-adic numbers coming as iterated p:th roots of starting point p-adic number one can sum each p-adic number separately. This is really cute if true!

It seems that one can formulate the arithmetics problem in the tilt where it becomes in principle as simple as in finite field with only p elements! Does the existence of solution in this case imply its existence in the case of p-adic numbers? But doesn't the situation remain the same concerning the existence of the solution in the case of rational numbers? The infinite series defining p-adic number must correspond a sequence in which binary digits repeat with some period to give a rational number: rational solution is like a periodic solution of a dynamical system whereas non-rational solution is like chaotic orbit having no periodicity? In the tilt one can also have solutions in which some iterated root of p appears: these cannot belong to rationals but to their extension by an iterated root of p.

The results of Scholze could be highly relevant for the number theoretic view about TGD in which octonionic generalization of arithematic geometry plays a key role since the points of space-time surface with coordinates in extension of rationals defining adele and also what I call cognitive representations determining the entire space-time surface if M⁸-H duality holds true (space-time surfaces would be analogous to roots of polynomials). Unfortunately, my technical skills in mathematics needed are hopelessly limited.

TGD inspires the question is whether the finite cutoffs of K_b - almost perfectoids - could be particularly interesting physically. At the limit of infinite dimension one would get an ideal situation not realizable physically if one believes that finite-dimensionality is basic property of extensions of p-adic numbers appearing in number theoretical quantum physics (they would related to cognitive representations in TGD). Adelic physics involves all extensions of rationals and the extensions of p-adic number fields induced by them and thus also extensions of type K_b. I have made some naive speculations about why just these extensions might be physically of a special signiticance.

See the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?. See also the article Could the precursors of perfectoids emerge in TGD?