# What is the analogy between the moduli of shtukas and Shimura varieties?

I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this is the case?

• There is a theorem (Berthelot, Gabber, ..) that says that the category of $p$-divisible groups over a perfect base scheme $R$ is equivalent to the category of "Dieudonne modules" over $W(R)$. These can be interpreted as Shtukas for the group GLn where the cocharacter $\mu$ is given by $(1,1,\cdots, 1, 0, \cdots, 0)$ where the number of 1's gives the dimension of the $p$-divisible group. – user45878 Mar 18 at 16:08
• Shimura varieties are moduli spaces of abelian varieties and map to these mixed-characteristic Shtukas roughly by the map $A \mapsto A[p^{\infty}]$ mapping an abelian variety to the associated $p$-divisible group. See Remark 5.2.2 of arxiv.org/abs/1707.05700. – user45878 Mar 18 at 16:20
• You have similar uniformization statements for Shimura varieties and shtukas (as double coset spaces such as $G(K)\backslash G(\mathbb{A})/G(\mathbb{O})$) which gives you a direct link to automorphic forms and tells you that in either case, the cohomology comes with a bunch of Hecke actions. – dhy Mar 20 at 2:25
• What is the uniformization statement for shtukas? – Kim Mar 20 at 7:52