Mathematical Research Letters

Volume 29 (2022)

Number 3

Perfectoid covers of abelian varieties

Pages: 631 – 662



Clifford Blakestad (Department of Mathematics, Pohang University of Science and Technology, Pohang, South Korea)

Damián Gvirtz (Department of Mathematics, University College London, United Kingdom)

Ben Heuer (Mathematical Institute, University of Bonn, Germany)

Daria Shchedrina (Altitude Gym, Kanata, Ontario, Canada)

Koji Shimizu (Department of Mathematics, University of California, Berkeley, Cal., U.S.A.)

Peter Wear (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Zijian Yao (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


For an abeloid variety $A$ over an algebraically closed non-archimedean field of residue characteristic $p$, we show that there exists a perfectoid space which is the tilde-limit of $\varprojlim_{[p]} \!\! A$. Our proof also works for the larger class of abeloid varieties.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC- 2047/1 – 390685813. During the preparation of this work, Clifford Blakestad was partially supported by NRF-2018R1A4A1023590. Dami´an Gvirtz and Ben Heuer were supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. Koji Shimizu was partially supported by NSF grant DMS-1638352 through membership at the Institute for Advanced Study. Peter Wear was supported by NSF grant DMS-1502651 and UCSD and would like to thank Kiran Kedlaya for helpful discussions. 660 C. Blakestad, et al. During the Arizona Winter School, Daria Shchedrina was supported by Peter Scholze and the DFG.

Received 29 September 2020

Accepted 22 January 2021

Published 30 November 2022