Having done some cursory searching of the internet, it is clear to me that there is a very well-developed theory of how to do calculus over function fields, such as fields of Laurent series in a single indeterminate with coefficients in a finite field (viz. Drinfeld modules, the Carlitz exponential, etc.) However, there seems to be nothing about the case where the function field has coefficients in a field of characteristic zero ($\mathbb{C}$, number fields, local fields, etc.). Is this because characteristic zero gunks up the theoretical machinery, or because no one has really explored it? Or do I just not know where to look for this?

References to articles/books about this topic (if they exist) would be much appreciated.

  • $\begingroup$ What is it you expect to see done when looking at "calculus over a function field"? Note that Laurent series over a finite field will be locally compact, while the x-adic topology of K((x)) is never locally compact if K is infinite (as all fields of characteristic 0 are). You might want to look up higher-dimensional local fields. $\endgroup$ – KConrad Apr 3 at 20:04
  • $\begingroup$ I'm particularly interested in analogues of certain features of the theory of analytic functions on $\mathbb{C}_{p}$, particularly the Shnirelman Integral formula, and a maximum-modulus property. Results about critical radii of power series would also be nice. $\endgroup$ – MCS Apr 3 at 21:44

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