Some experiments investigating a possible L(1/4) algorithm for the discrete logarithm problem in algebraic curves

The function field sieve, a subexponential algorithm of complexity L(1/3) that computes discrete logarithms in finite fields, has recently been improved to an algorithm of complexity L(1/4) and subsequently to a quasi-polynomial time algorithm. We investigate whether the new ideas also apply to index calculus algorithms for computing discrete logarithms in Jacobians of algebraic curves. While we do not give a final answer to the question, we discuss a number of ideas, experiments, and possible conclusions