Which languages could appear on Weil's Rosetta Stone?

Question

André Weil's likening his research to the quest to decipher the Rosetta Stone (see this letter to his sister) continues to inspire contemporary mathematicians, such as Edward Frenkel in Gauge Theory and Langlands Duality.

Remember that Weil's three 'languages' were: the 'Riemannian' theory of algebraic functions; the 'Galoisian' theory of algebraic functions over a Galois field; the 'arithmetic' theory of algebraic numbers. His rationale was the desire to bridge the gap between the arithmetic and the Riemannian, using the 'Galoisian' curve-over-finite-field column as the best intermediary, so as to transfer constructions from one side to the other. (See also 'De la métaphysique aux mathématiques' 1960, in volume II of his Collected Works.)

That fitted rather neatly with demotic Egyptian mediating between priestly Egyptian (hieroglyphs) and ordinary Greek on the real Rosetta Stone. But just as one might have expressed that text in a range of other contemporary languages - Sanskrit, Aramaic, Old Latin, why should there not be other columns in Weil's story? Frankel himself adds a fourth column (p. 11) 'Quantum Physics'.

So now the questions:

Are there other candidate languages for Weil's stone? Might there be a further language for which we would need intermediaries back to the arithmetic? Could there be a meta-viewpoint which determines all possible such languages.

Presumably the possession of a zeta function is too weak a condition as that would allow the language of dynamical systems.

Answer 1

All the 12 or more approaches to geometry over the field with one element are tentatives to create such intermediate languages. But you seemed to ask more about a pre-existing area of it's own which may serve as a bridge - in this direction there are

$\bullet$ Alexandru Buium's theory of Arithmetic Differential Equations which brings the theory of differential equations into play as an intermediate language.

$\bullet$ I remember Alexandru Buium saying that there is also a theory of difference equations, different from his, but don't know more about this.

$\bullet$ Shai Haran uses probability theory as an intermediate language in his book "Mysteries of the Real Prime". It connects to the quantum theory column, but I don't know whether it's in the same way that Frenkel suggested.

$\bullet$ Homotopy theory might be a future candidate for a column of its own, on the one hand via motivic homotopy theory as it is getting available over more and more general base schemes with more and more general coefficients and thus moving towards arithmetic. And on the other hand possibly via ring spectra which may serve as the base deeper than the integers which is hoped to trigger the translation process one day...

$\bullet$ You write that the possession of Zeta functions is too weak to make the theory of dynamical systems an intermediate language. But there is certainly much more connecting dynamical systems with Arithmetic, as shown in Deninger's work.

$\bullet$ The works of Bost, Connes, Marcolli, Meyer, Laca and others connect Arithmetic to the Theory of Operator Algebras, and via those again to dynamical systems, quantum physics and Thermodynamics

Answer 2

As was said in the comments this question is a bit open ended, but could lead to some cool answers.

Something I personally find very cool along these lines is Vojta's dictionary between Nevanlinna theory (a branch of complex function theory) and Diophantine approximation.

Serge Lang wrote a book on Nevanlinna theory for these reasons.

Yamanoi has made some recent progress along these lines.