Zagier’s paper introduced in the first half of the working group had elementary proofs which benefits the readers in a way of being easily understood. However, it has a drawback of the lack of motivation for the construction. To improve or to simply replicate the result, one needs to understand the geometric construction behind this paper. Unfortunately, this requires the audience to assume many notions from algebraic geometry. After directly giving these notions to the audience, I will define the general notion of height functions. In fact, by analytifying the algebraic variety, one could easily derive a non-trivial lower bound of heights. And I will show how to identify this general notion of height functions with the special case of projective height and Zhang-Zagier height. The general lower bound motivates the notion of successive minimums and essential minimums. Finally, we will give a procedure of how to produce or to approximate successive minimums, and we will specify how Zagier utilized the idea of this procedure for the construction of this paper. No number theory and algebraic geometry background is needed. These two talks will be supplemented by a series of notes I’ve been writing, which will be sent by the organizer after the talk.