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Local Fourier transforms and rigidity for \({\mathcal D}\)-modules. (English) Zbl 1082.14506
From the text: In an important article, G. Laumon [Publ. Math., Inst. Hautes Étud. Sci. 65, 131–210 (1987; Zbl 0641.14009)] applied the \(\ell\)-adic Fourier transform to study epsilon factors associated to \(\ell\)-adic sheaves on curves over finite fields. As a key tool, he defined local Fourier transforms \(\mathcal F(0,\infty)\), \(\mathcal F(\infty,0)\), \(\mathcal F(\infty,\infty)\) for \(\ell\)-adic sheaves on \(\text{Spec}\,\mathbb F_q((t))\). The purpose of this paper is to develop local Fourier transforms for meromorphic connections over Laurent series fields \(k((t))\). We show that these have properties precisely analogous to Laumon’s local \(\ell\)-adic local Fourier transforms.
Following a program of N. M. Katz [Rigid local systems, Ann. Math. Stud. 139 (1969; Zbl 0864.14013)], 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 conditionis satisfied. As in Katz’ book, 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 [P. Deligne, Equations différentielles à point singulier, Lecture Notes Math. 163, Springer, Berlin (1970; Zbl 0244.14004)].

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
11G99 Arithmetic algebraic geometry (Diophantine geometry)
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