Cohomology Classes from q-Picard–Fuchs Equations

In joint work with Garoufalidis–Scholze–Zagier, we constructed elements in generalized Habiro rings associated to number fields. A central aspect of this work involved lifting analytic to algebraic data—a process that can be viewed as a form of analytic continuation. In ongoing work with Garoufalidis, we extend these techniques to the setting of hypergeometric motives, replacing algebraic functions with periods. When interpreted appropriately, this yields power series expansions around roots of unity whose coefficients are explicitly computable cohomology classes. We anticipate that these constructions will produce concrete examples of classes in the Habiro cohomology framework recently developed by Wagner and Scholze.