We explain how our joint work with Frank Calegari and Yunqing Tang further refines the unbounded denominators conjecture to take the form of a converse theorem for GL(2,A_Q) without any twists by Dirichlet characters. The thrust of such theorems is to replace Weil's character twists by a bounded denominators condition on the Dirichlet series coefficients. This opens up a view towards arithmetic algebraization methods for Dirichlet series, and motivates to search for integral converse statements for more general groups. We will conclude by discussing the situation for Hilbert modular groups.