An explicit stationary phase formula for the local formal Fourier-Laplace transform

We give an explicit formula (i.e., an explicit expression for the formal stationary phase formula) for the local Fourier-Laplace transform of a formal germ of meromorphic connection of one complex variable with a possibly irregular singularity. Introduction We will denote by C((t)) (resp. C({t})) the field C[[t]][t] (resp. C{t}[t]) of formal (resp. convergent) Laurent series of the variable t, equipped with its usual derivation ∂t. Let M be a finite dimensional C((t))-vector space with a connection ∇. The local formal Laplace transform F (0,∞) (also called Fourier transform in the literature) was introduced in [1, 4] by analogy with the l-adic local Fourier transform considered in [6]. One way to produce it is to choose a free C[t, t]-module M of finite rank equipped with a connection ∇ having poles at most at t = 0 and t = ∞, with a regular singularity at infinity, and such that (C((t))⊗C[t,t−1] M , 1⊗∇) = (M,∇). Considering M as a C[t]〈∂t〉-module, its (global) Laplace transform FM is the same C-vector space equipped with the C[τ ]〈∂τ 〉-structure defined by the correspondence τ = ∂t, ∂τ = −t. Tensoring with C[τ, τ ] gives a C[τ, τ]〈τ∂τ 〉-module FM [τ], and renaming θ = τ (and setting θ∂θ = −τ∂τ ), we regard FM [τ ] as a C[θ, θ]〈θ∂θ〉-module. Lastly, we define F M as C((θ))⊗C[θ,θ−1]FM [τ ], equipped with its natural connection. This does not depend of the choices made. The previous transform corresponds to using the kernel e, and is also denoted by F (0,∞) − . Its inverse transform is denoted by F (∞,0) + . There are also pairs of inverse transforms (F (s,∞) ± ,F (∞,s) ∓ ) for any s ∈ C and (F (∞,∞) ± ,F (∞,∞) ∓ ). If we denote by F± the algebraic Laplace transform with kernel e ±tτ acting on a C[t]〈∂t〉module M , the local formal stationary phase formula of [1, 4] relies the formalization of the Laplace transform of M at each of its singularities (0, ŝ ∈ C, ∞̂) with the local formal Laplace transforms of M itself at its singularities 0, s ∈ C, ∞. 2000 Mathematics Subject Classification. Primary 32S40; Secondary 14C30, 34Mxx.