Langlands correspondence for isocrystals and the existence of crystalline companions for curves

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<p>In this paper, we show the Langlands correspondence for isocrystals on curves, which asserts the existence of crystalline companions in the case of curves. For the proof we generalize the theory of arithmetic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-modules to algebraic stacks whose diagonal morphisms are finite. Finally, combining with methods of Deligne and Drinfeld, we show the existence of an “<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic companion” for any isocrystal on a smooth scheme of any dimension under the assumption of a Bertini-type conjecture.</p>

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