nLab shtuka




In arithmetic geometry over a finite field a shtuka on an arithmetic scheme is essentially an equivariant algebraic vector bundle on the product of the scheme with a given arithmetic curve, where equivariance is with respect to the action of the Frobenius endomorphism (e.g. Scholze-Weinstein, def. 1).

(Shtuka is a Russian word colloquially meaning “thing”.)



(Definition 7.1 in #Scholze17)

Let XX be a smooth projective curve over 𝔽 p\mathbb{F}_{p} and let SS be a scheme over 𝔽 p\mathbb{F}_{p}. A shtuka over SS relative to XX with legs at x 1,,x n:SXx_{1},\ldots,x_{n}:S\to X is a vector bundle \mathcal{E} over S× 𝔽 pXS\times_{\mathbb{F}_{p}}X together with an isomorphism

Frob S *| S× 𝔽 pX i=1 nΓ x i| S× 𝔽 pX i=1 nΓ x i\mathrm{Frob}_{S}^{\ast}\mathcal{E}\vert_{S\times_{\mathbb{F}_{p}} X\setminus \bigcup_{i=1}^{n}\Gamma_{x_{i}}}\cong\mathcal{E}\vert_{S\times_{\mathbb{F}_{p}} X\setminus \bigcup_{i=1}^{n}\Gamma_{x_{i}}}


Reviews of the basic definition include

Definition 1 in

More conceptual discussion, in the context of the function field analogy, is in

Last revised on February 15, 2024 at 16:05:06. See the history of this page for a list of all contributions to it.