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”.)

Related concepts

• Rapoport-Zink space

References

Reviews of the basic definition include

• David Goss, What is … a Shtuka?, Notices of the AMS 50(1), pp. 36-37.

• 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)