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We give an introductory account of two recent approaches towards an effectiveproof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The lattermethod, which is usually called the method of Chabauty--Kim or non-abelianChabauty in the literature, has the advantage that in some cases it has beenturned into an effective method to determine the set of rational points on acurve, and we illustrate this by presenting three new examples of modularcurves where this set can be determined.

Two recent p-adic approaches towards the (effective) Mordell conjecture