Microlocalization and stationary phase

Details Microlocalization and stationary phase Microlocalization and stationary phase

2003-07-28

arxiv.org/abs/math/0307366v1

Mathematics

Algebraic Geometry

32 pages

Scientific paper

Let M be a holonomic module over the Weyl algebra K[t]<\partial_t>, K a field of characteristic zero. We prove a stationary phase formula which expresses the formalization of the germ at infinity of the Fourier transform of M in terms of a sum of local contributions depending on the germs defined by M at its singular points and at infinity. For this purpose, we consider formal analogues of the local Fourier transforms defined by G. Laumon in the l-adic setting (for instance, the transformation labelled (0,\infty) by Laumon corresponds in our context to formal microlocalization). When K is the field of complex numbers we can describe in a similar way the 1-Gevrey germ at infinity defined by M. When K is a p-adic field, we make a modest attempt to reproduce a small part of these constructions in the p-adic setting. We define a ring of p-adic microdifferential operators (of finite order) and we prove a p-adic stationary phase formula in some special cases.

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