Considering a holonomic ${\mathcal D}$-module and a hypersurface, we define a finite family of ${\mathcal D}$-modules on the hypersurface which we call modules of vanishing cycles. The first one had been previously defined and corresponds to formal solutions. The last one corresponds, via Riemann-Hilbert, to the geometric vanishing cycles of Grothendieck-Deligne. For regular holonomic ${\mathcal D}$-modules there is only one sheaf and for non regular modules the sheaves of vanishing cycles control the growth and the index of solutions. Our results extend to non holonomic modules under some hypothesis.

We give a generic estimation of trigonometric sums defined over closed sub-schemes with semi-stable reduction of the standard affine scheme modulo pn(n [ges ] 2). We use Greenberg realisation to reduce to trigonometric sums defined over smooths sub-schemes of a finite product of Witt vectors over the finite field of p elements. Using the cohomological interpretation of this sums over a finite field, the sum is directly related to the Fourier–Deligne transformation of the dual pairs of Witt vectors. We deduce the estimation from the properties of the Fourier–Deligne transformation on simple perverse sheaves and pure sheaves.

We prove that a refinement of Stark‘s Conjecture formulated by Rubin in Ann. Inst Fourier 4 (1996) is true up to primes dividing the order of the Galois group, for finite, Abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions, a statement stronger than Rubin‘s holds true.