transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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 $X$ 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
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
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
Last revised on February 15, 2024 at 16:05:06. See the history of this page for a list of all contributions to it.