On Galois action in rigid DAHA modules

Given an elliptic curve over a field $K$ of algebraic numbers, we associate with it an action of the absolute Galois group $G_K$ in the type $A_1$ rigid DAHA-modules at roots of unity $q$ and over the rings $Z[q^{1/4}]/(p^m)$ for sufficiently general prime $p$. We describe rigid modules in characteristic zero and for such rings. The main examples of rigid modules are generalized nonsymmetric Verlinde algebras; their deformations for arbitrary $q$ are constructed in this paper, which is of independent interest on its own. The Galois action preserves the images of the elliptic braid group in the groups of automorphisms of rigid modules over $Z[q^{1/4}]/(p^m)$. If they are finite in characteristic zero, then $G_K$ acts there and no reduction modulo $(p^m)$ is needed; we find all such cases. In the case of $3$-dimensional DAHA-modules, these images are quotients of equilateral triangle groups directly related to the Livn\'e groups. Also, this paper can be viewed as an extension of the DAHA theory of refined Jones polynomials of torus knots (for $A_1$) to $G_K$.