  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 $G_K$.
