Theory Of Fourier Transform For D-modules

In mathematics,Fourier analysis and Fourier transform have been developed for a long time.Fourier analysis and Fourier transform have many scientific applications, such as in physics,partial differential equations,number theory,cryptography, numerical analysis,optics,geometry,and other areas.The stationary phase approximation is a basic principle of a special class of asymptotic fourier analysis, such as oscillatory integrals.Under some conditions,the major contribution to the value of oscillatory integral comes from neighborhoods of those critical points of the integrand.The theory of local Fourier transforms in cohomological theory was first introduced by Laumon[50],in order to study the local behavior of global l-adic Fourier transform which was first introduced by Deligne.Laumon[50]discovered the stationary phase principle.The global Fourier transform of a D-module on the affine line is defined to be the the Gauss-Manin connection on some relative de Rham cohomology. The global Fourier transform can also be obtained by switching(up to sign) the multiplication by a coordinate of the affine line and the differential operator with respect to this coordinate.The theory of local Fourier transforms for D-modules were respectively introduced by Lopez[52]and by Bloch and Esnault[4].In[52],Lopez defined local Fourier transforms of holonomicκ[t,(?)_t]-modules using rings of formal microdifferential operators.Lopez proved a stationary phase formula expressing the formal germ at infinity of the Fourier transform of a holonomicκ[t,(?)_t]-module M in terms of the formal germs of M at its singular points and at infinity.Local Fourier transforms can be viewed as the restriction of global Fourier transform.From this point of view,Bloch and Esnault[4]defined the local Fourier transforms by the technical tool of good lattices pairs of connections.Using this method,they proved a stationary phase formula.That is,the formal germ at infinity of the Fourier transform of a vector bundle with a connection on an open dense subscheme of the affine line,is the direct sum of local Fourier transforms of the vector bundle from its singular points to infinity.However,there were many errors in the paper.On the basis of these background,my doctoral dissertation is mainly devoted to completing the theory of local Fourier transforms for D-modules.The dissertation is organized as follows. In chapter 2,we mainly discuss good lattices pairs of connections.Particularly, we discuss good lattices pairs of connections on a discrete valuation field. For a connection on a field of formal Laurent series with pure slope,we define a valuation on this connection such that the differential operator is pure.In chapter 3,we define the global Fourier transform of a vector bundle with a connection on an open dense subscheme of the affine line,by the Guass-Manin connection on some relative de Rham cohomology.Choose a suitable good lattices pair of the vector bundle,we rewrite the relative de Rham cohomology by coherent modules.Using Grothendieck's proper base change theorem,we prove the stationary phase formula.In chapter 4,we study properties of local Fourier transforms.In the last part of this chapter,we compare these definitions of local Fourier transforms with that defined by Lopez.In chapter 5,we gives explicit formulas for local Fourier transforms from 0 to infinity,form infinity to 0,and from infinity to infinity,respectively.It indicates the relation between local Fourier transforms and Legendre transforms.