I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up the calculation.  

My question is, how is computation of the rank of an elliptic curve made faster by assuming the GRH. I am no expert in this field, so please keep your answers as simple as possible.



GRH and the rank of elliptic curves
Computation of ranks of elliptic curves relies on descent. The first step of descent is the computation of a finite Selmer group, which in turn uses the computation of the class group of a potentially large number field. This is the step where GRH is used: it allows you to assume that the class group is generated by the set of prime ideals up to a relatively small norm bound, therefore speeding up the computation.


