# Exponential sums and differential equations

Hi, I have a general question about the relationship between exponential sums and differential equations. In particular, I have been trying to read Katz' work on the subject (his book and his lecture notes) but I am having trouble understanding the big idea and getting confused with all of the algebraic geometry background. Can someone explain the general picture, or direct me towards more elementary works introducing the subject?

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A motivation of the book of N. Katz is study of the monodromy of hypergeometric differential equations. Good pre-references could be Levelt's 1961 PhD thesis "Hypergeometric functions" and [F. Beukers and G. Heckman, Monodromy for the hypergeometric function $_nF_{n-1}$, Invent. Math. 95 (1989) 325--354; dx.doi.org/10.1007/BF01393900]. – Wadim Zudilin Oct 10 '10 at 22:25

## 1 Answer

In simplistic terms, exponential sums arise in characteristic p geometry because curves have Artin-Schreier coverings. The appearance of (complete) exponential sums in that context is a reflection in cohomology of that kind of Galois covering, and can be understood pretty much in Weil's terms and technology. (Appendix in Basic Number Theory? Unfortunately I have the first edition that doesn't have such conveniences.) And linear differential equations on curves have been understood geometrically since Riemann's time, admittedly in varying languages. The case corresponding to finite coverings (i.e. curve morphisms) is that of finite monodromy.

So far so good? Nick Katz's papers have got easier to read as the years go by (at least 10% per decade I think, as he has moved away from the Grothendieckising style). But are probably still hard going.

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