In joint work with Gareth Jones we prove an effective version of a result of the speaker with Daniel Bertrand. This involves proving an effective and uniform version of a theorem of Pila and Wilkie on counting rational points in o-minimal structures that is valid in this setting. Another ingredient is a height inequality that allows one to prove an effective version of the Mordell-Lang conjecture under some circumstances. 

I will give a gentle introduction to generalised Jacobians before motivating the result and providing some applications. Time permitting, I will get into some of the technical difficulties of the proof.

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