# Moduli spaces of shtukas

# Moduli spaces of shtukas

This chapter examines the moduli spaces of mixed-characteristic local *G*-shtukas and shows that they are representable by locally spatial diamonds. These will be the mixed-characteristic local analogues of the moduli spaces of global equal-characteristic shtukas introduced by Varshavsky. It may be helpful to briefly review the construction in the latter setting. The ingredients are a smooth projective geometrically connected curve *X* defined over a finite field **F**q and a reductive group *G/ Fq*. Each connected component is a quotient of a quasi-projective scheme by a finite group. From there, it is possible to add level structures to the spaces of shtukas, to obtain a tower of moduli spaces admitting an action of the adelic group. The cohomology of these towers of moduli spaces is the primary means by which V. Lafforgue constructs the “automorphic to Galois” direction of the Langlands correspondence for

*G*over

*F*.

*Keywords:*
moduli spaces, shtukas, spatial diamonds, mixed-characteristic local analogues, Varshavsky, quasi-projective scheme, adelic group, V. Lafforgue, Langlands correspondence

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