Shtuka - Knowlpedia

Let $X$ be a smooth projective curve over the finite field $\mathbb F_q$ , let $S$ be an $\mathbb F_q$ -scheme, and let $x_i:S\to X$ be finitely many legs. In the multiple-leg form, a rank- $n$ shtuka consists of an algebraic vector bundle $\mathcal E$ on $X\times S$ and an isomorphism

\varphi: \mathcal E\big|_{(X\times S)\setminus\bigcup_i\Gamma_{x_i}} \xrightarrow{\ \sim\ } ({\rm id}_X\times{\rm Frob}_S)^*\mathcal E \big|_{(X\times S)\setminus\bigcup_i\Gamma_{x_i}},

with prescribed bounds on the relative positions at the graphs $\Gamma_{x_i}$ . Here ${\rm Frob}_S$ denotes the absolute Frobenius endomorphism of $S$ .

Frobenius bundle with controlled singularities

Without legs, $\varphi$ is a global Frobenius descent datum. The legs allow $\varphi$ to have controlled zeros or poles, expressed invariantly as modifications of vector bundles. Classical Drinfeld shtukas often have two distinguished legs, customarily called a zero and a pole.

The rank- $n$ definition is the $\operatorname{GL}_n$ -case of a $G$ -shtuka.

Moduli interpretation

Allowing the legs and bundle to vary gives an algebraic stack over $X^I$ . Bounds by dominant coweights produce finite-type truncations after imposing level and Harder–Narasimhan conditions. The fibers carry actions of Hecke correspondences and partial Frobenius morphisms.

Arithmetic role

Drinfeld used shtukas to prove the global Langlands correspondence for $\operatorname{GL}_2$ over function fields; Laurent Lafforgue extended this to $\operatorname{GL}_n$ . Vincent Lafforgue's construction for general reductive groups uses multiple-leg $G$ -shtukas, the geometric Satake equivalence, and excursion operators.

Not the same as a local shtuka

A local shtuka in the Fargues–Fontaine setting is a local modification object over a nonarchimedean local field. It is inspired by the same Frobenius-and-modification pattern but lives in a different geometry.

References

V. G. Drinfeld, “Moduli varieties of $F$ -sheaves,” Functional Analysis and Its Applications 21 (1987), 107–122. DOI.
Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” 2018. arXiv.