863: Structural Mapping Theory/1

[Timothy Y. Chow]

Harvey Friedman wrote:

> I am tentatively calling this new field "structural mspping theory". 
> abbreviated SMAT.
>
> GENERAL QUESTION. GIven two algebraic systems, is there a mapping from 
> the first into the second which preserves the solvability of finite 
> systems of equations and is not surjective? Or preserves a variety of 
> fundamental conditions and is not surjective?

This is vaguely reminiscent of zeta functions of varieties over finite 
fields.  Although it's only a family resemblance, let me say a bit more 
since I know you're always on the lookout for connections with "mainstream 
mathematics."

At some early point in our mathematical education, we are taught that a 
curve (or its higher-dimensional generalization, a variety) is a set of 
points.  But in modern algebraic geometry, a curve is a more abstract 
object, and if we want to talk about a "set of points" then we need to 
choose a field F and speak of the "F-rational points" of the curve.

If we have a curve defined over a finite field F_q (which roughly speaking 
means we have written down a finite set of equations with coefficients in 
F_q) then we can let N_k be the number of F_{q^k}-rational points for 
every integer k >= 1.  The numbers N_k can be collected together into a 
generating function called the (local) zeta function of the curve.

https://en.wikipedia.org/wiki/Local_zeta-function

The zeta function, more or less, tracks "the solvability of a finite 
system of equations."  I say "more or less"; it does more, in the sense 
that it tracks not just whether the system is solvable but how many 
solutions it has (tracking infinitely many fields at once), and it does 
less, in the sense that a curve (or a variety) isn't *exactly* the same as 
a finite system of equations.

One can then ask about maps between varieties that preserve the zeta 
function.  I don't know much about this topic.  Certainly different 
varieties can have the same zeta function.  I found one MathOverflow 
question that seems to be relevant but that only scratches the surface.

https://mathoverflow.net/q/19802

With a lot more work, one can also define the Hasse-Weil zeta function of 
a number field (a number field is a finite algebraic extension of the 
rationals).  Roughly speaking, the Hasse-Weil zeta function combines 
infinitely many local zeta functions, one for each prime.  Two number 
fields with the same Hasse-Weil zeta function are called "arithmetically 
equivalent" and, as you might expect, necessarily share a lot of 
properties, but not everything of interest (e.g., they may have different 
class numbers).

Tim