This is one of the many examples of geometric concepts being applied to integers (in this case, the notion of derivation, although a non-linear one).
Another important concept is that of a curve and its ring of functions in algebraic geometry; for the integers, the curve is the prime spectrum of Z, i.e. the prime ideals generated by each prime number <p>. The ring of regular functions is precisely the ring of integers, operating as functions on prime numbers by n(p) = n modulo p.
I wonder if D has any interpretation in terms of nonlinear differential operators on Spec(Z).
Ahh, nice one dares to proffer this. I can plug my own wondering about if one can formulate an aritmetic derivative that instantiates the Kähler differential concept (but I'm supposed not to ask unless I already knew the answer, so what'd'be the point).
Are you saying the derivative is a geometric concept? Tangent slope of a curve is simply one application of a derivative; it's not the derivative's identity. What the derivative is is the inverse of an inner product.
Can you people stop with the inane pedantry? Yes, the derivative is a geometric concept and so is the inner product; they are at the core of what a Riemannian manifold is, they group to form the (co)tangent spaces of varieties and schemes and their derived structures produce the local geometric data of the object in question.
The point is that the derivative is a more general concept than just geometric, and is naturally defined with or without a geometric comtext. Of course almost anything can be modeled geometrically. You can draw a picture of almost anything. The integers themselves are obviously geometric, by drawing a kindergarten number line.
This is all semantic nonsense that devalues the original post I made about a potential connection between two arithmetic-geometric objects. But thanks for patronizing me, a research mathematician actually working on the arithmetic Langlands program, with a wikipedia link to the fucking Langlands program.
Another important concept is that of a curve and its ring of functions in algebraic geometry; for the integers, the curve is the prime spectrum of Z, i.e. the prime ideals generated by each prime number <p>. The ring of regular functions is precisely the ring of integers, operating as functions on prime numbers by n(p) = n modulo p.
I wonder if D has any interpretation in terms of nonlinear differential operators on Spec(Z).