Local Fourier Transforms and Rigidity for D-Modules

Local Fourier transforms, analogous to the ℓ-adic local Fourier transforms [14], are constructed for connections over k((t)).Following a program of Katz [12], a meromorphic connection on a curve is shown to be rigid, i.e. determined by local data at the singularities, if and only if a certain infinitesimal rigidity condition is satisfied.As in [12], the argument uses local Fourier transforms to prove an invariance result for the rigidity index under global Fourier transform.A key technical tool is the notion of good lattice pairs for a connection [5].