nLab shtuka

Idea

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

Definition

(Definition 7.1 in #Scholze17)

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

$$\mathrm{Frob}_{S}^{\ast}\mathcal{E}\vert_{S\times_{\mathbb{F}_{p}} X\setminus \bigcup_{i=1}^{n}\Gamma_{x_{i}}}\cong\mathcal{E}\vert_{S\times_{\mathbb{F}_{p}} X\setminus \bigcup_{i=1}^{n}\Gamma_{x_{i}}}$$

Related concepts

• Rapoport-Zink space

References

Reviews of the basic definition include

• David Goss, What is … a Shtuka?, Notices of the AMS 50 1 (2003) 36-37 [pdf, full issue:pdf]

• Wikipedia, Drinfeld module – Shtuka

Definition 1 in

• Jared Weinstein, notes from lecture by Peter Scholze, Peter Scholze’s lectures on $p$-adic geometry, MSRI 2014 (pdf)

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

• Urs Hartl, A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic (arXiv:math/0607182)

• Peter Scholze, p-adic geometry, arXiv:1712.03708