Wojciech J. Gajda

Let f(x), g(x) be rational functions with complex coefficients. How does one determine the intersection of the subfields C(f) and C(g) inside C(x)? More generally, let X <- Y -> W be branched covers between Riemann surfaces. Are there covering maps from X, Y, W to a surface P, making the diagram commutative? If that is the case, we say that the correspondence fits in a Galois diagram. This seemingly elementary problem plays a central role in understanding how Jacobians decompose into simple components, up to isogeny. Suppose Y -> X is a ramified cover of algebraic curves such that there exists a curve W for which JacY ~ JacX x JacW. A natural question is how to describe W. A guiding conjecture predicts that W can be identified using Galois-theoretic methods applied to the covering Y -> X. It is equivalent to asking whether a certain correspondence X <- Y* -> W fits in a Galois diagram.