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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)].

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