Local shtuka - Knowlpedia

In the Fargues–Fontaine formulation, a local $G$ -shtuka is a Frobenius-compatible modification of $G$ -bundles at one or more untilt divisors, with relative positions bounded by specified conjugacy classes of cocharacters.

A typical moduli space

\operatorname{Sht}(G,b,\mu_\bullet)_K

parametrizes a modification from the bundle $\mathcal E_b$ to the trivial $G$ -bundle, bounded at its legs by $\mu_\bullet=(\mu_i)$ , together with a $K$ -level trivialization, where $K\subset G(E)$ is compact open.

Geometric data

Equivalently, on the Frobenius cover $Y_{S,E}$ , one describes a $G$ -torsor with an isomorphism to its Frobenius pullback away from the leg divisors. Quotienting by Frobenius translates this into modifications of bundles on the Fargues–Fontaine curve.

Multiple legs and ordered modifications give convolution versions. A period map sends the moduli space to a twisted affine Grassmannian bounded by the cocharacters.

Group actions

The tower over varying $K$ carries commuting actions associated to $G(E)$ , the self-quasi-isogeny group $G_b(E)$ , and Weil groups of the reflex fields of the legs. Its compactly supported cohomology therefore produces representations on both the automorphic and Galois sides.

Special cases

For one minuscule leg and suitable local Shimura data, these spaces recover local Shimura varieties, including Rapoport–Zink spaces. General local shtuka spaces allow arbitrary cocharacter bounds and several legs.

Terminology boundary

Equal-characteristic local $G$ -shtukas can also be defined using loop groups over formal discs. The displayed Fargues–Fontaine definition is the mixed/equal-characteristic geometric framework used by Fargues–Scholze; it should not be confused with a global $G$ -shtuka on a projective curve.

References

Peter Scholze and Jared Weinstein, Berkeley Lectures on $p$ -adic Geometry, Chapter 23, 2020. AMS.
Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” §§I.7 and IX.3. arXiv.