On the infinite Frobenius actions on de Rham fundamental groups of affine curves

We compute the infinite Frobenius actions on the de Rham fundamental groups of affine curves defined over R. As an application, we compute extension classes of real mixed Hodge structures associated with the motivic fundamental groups of affine curves. We establish connections between our computation and special values of Rankin-Selberg L-functions, focusing on the case of modular curves, and show that the associated extensions of mixed Hodge structures does not split. We compute local zeta integrals not only at good primes but also at certain bad primes.