2019-06-14 00:07:40 8 Comments

Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper

Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–1192, doi:10.5802/afst.1563, arXiv:1606.01796

Especially intriguing to me is Conjecture 3.3: Betti-de Rham comparison isomorphism.

In the paper the author emphasized the lack of nontrivial examples and suggests to do a computation for the universal family of elliptic curves.

**Question 1:** Has anyone done this computation (or any other interesting computations)? I will be especially interested in those involving basic hypergeometric functions.

**Question 2:** What does Conjecture 3.3 mean in practice? Naively one might hope that it leads to some $q$-deformation of periods.

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