Publications

Abstract: In 1981, Mukai constructed the Fourier–Mukai transform for abelian varieties over an algebraically closed field, which gives an equivalence of categories between quasi-coherent sheaves over A and the one over A∨, its dual variety. Laumon generalized these results for abelian varieties over a locally noetherian base. In this article, we define a Fourier–Mukai transform for an abelian formal scheme A/S = Spf(V), where V is a discrete valuation ring, and we extend the classical results of Fourier–Mukai transform to this case. Finally, we discuss the case of the generic fiber AK of A to obtain an equivalence of categories between coherent sheaves over AK and the ones over A∨K 

Abstract: In 1996, Rothstein and Laumon simultaneously constructed a Fourier-Mukai transform for D-modules over a locally noetherian base of characteristic 0. This functor induces an equivalence of categories between quasi-coherent sheaves of D-modules over an abelian variety A and quasicoherent sheaves of O-modules over the universal vectorial extension of its dual abelian variety. In this article, we define a Fourier-Mukai transform for D-modules on an abelian formal scheme A/S = Spf (V), where V is a discrete valuation ring, and we discuss the extension of the classical results of Fourier-Mukai transform to this arithmetic case.