» Katz N.M. - Exponential sums and differential equations
Katz N.M. - Exponential sums and differential equations
||Exponential sums and differential equations
||such a way that the two complement each other, with the applications serving both to illustrate and to motivate the general theory. The reader will judge how well we have succeeded.
Parts II and III especially are written in this spirit of "applied mathematics"; there is a strong emphasis on the effective calculation of the groups Ggai and G_eom respectively, when one is
given a concrete differential equation, or a concrete one-parameter family of exponential sums.
The effective calculations in Parts II and III ultimately rely on the general representation-theoretic results of Part I. However, in order to be able to bring these results to bear, we make essential use of some recent developments in the theory of differential equations and in the theory of one-parameter families of exponential sums.
In the case of differential equations, there are four essential ingredients. The first is the theory of the "slopes", or "breaks", of a differential equation on an open curve at one of the points at infinity. The second is the general theory of holonomic JD- modules on curves, especially the theory of the "middle extension" and the structure theory of irreducible J)-modules. The third is the theory of the Fourier
Transform of JD-modules on A'. The fourth is the idea of the (derived category) convolution of holonomic J)-modules on both the additive group A^ and on the multiplicative group 6m. [There is also a natural
notion of convolution of holonomic JD-modules on elliptic curves, which seems well worth exploring.]
What happens in the case of one-parameter families of exponential sums? Roughly speaking, studying a "one parameter family" means studying a lisse fc-adic sheaf on an open curve over a field of postitve characteristc p * I. In this case as well there are four essential ingredients, which are closely analogous to the JD-module ingredients discussed above. The first is the the theory of "breaks" (in the sense of the upper-numbering filtration) of t-adic representations of inertia groups at the points at infinity. This theory was in fact the inspiration for its D.E. namesake. The second is the theory of perverse i-adic sheaves on curves, especially the structure theory of irreducibles. This theory is analogous to the theory of holonomic J)-modules on curves over C. The third is the theory of the t-adic Fourier Transform for perverse sheaves on A' over a field of positive characteristic p * t. In the t-adic case, we have much more precise information about Fourier Transform than we do in the I)-module case, thanks to Laumon's "principle of stationary phase", which