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

We study the action of the infinite Frobenius on the de Rham fundamental groups of affine curves defined over $\text{bfR}$. As an application, we compute extension classes of real mixed Hodge structures associated with the motivic fundamental groups of affine curves. In the case of modular curves, we relate our computation to special values of Rankin-Selberg L-functions, and show that the associated extensions of mixed Hodge structures are non-split. We compute local zeta integrals both at good primes and, in certain cases, at bad primes.