A new approach has been recently developed to study the arithmetic of hyperelliptic curves 𝑦 2 = 𝑓(𝑥) over local fields of odd residue characteristic via combinatorial data associated to the roots of 𝑓. Since its introduction, numerous papers have used this machinery of 'cluster pictures' to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.

Abstract. We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the p-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the form y 2 = f (x), under some simplifying hypotheses.

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We develop an effective version of the Chabauty–Kim method which gives explicit upper bounds on the number of $S$ -integral points on a hyperbolic curve in terms of dimensions of certain Bloch–Kato Selmer groups. Using this, we give a new ‘motivic’ proof that the number of solutions to the $S$ -unit equation is bounded uniformly in terms of $\#S$ .