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Let X be a curve of genus g > 1 over Q whose Jacobian J has Mordell-Weil rank r and Néron-Severi rank ρ. When r < g + ρ − 1, the geometric quadratic Chabauty method determines a finite set of p-adic points containing the rational points of X. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of p-adic heights and p-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of p-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference.

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Given a finite abelian group G and a natural number t, there are two natural substructures of the Cartesian power G t ; namely, S t where S is a subset of G, and x+H a coset of a subgroup H of G t . A natural question is whether two such different structures have non-empty intersection. This turns out to be an NP-complete problem. If we fix G and S, then the problem is in P if S is a coset in G or if S is empty, and NP-complete otherwise; if we restrict to intersecting powers of S with subgroups, the problem is in P if n∈Z|nS⊂S nS is a coset or empty, and NP-complete otherwise. These theorems have applications in the article [Spe21], where they are used as a stepping stone between a purely combinatorial and a purely algebraic problem.Remark 1.4. Note that R, G, S are not part of the input of the problem. In particular, computations inside G can be done in O(1).Note these problems are certainly in NP, as one can easily give an R-linear combination of the generators (and add x * if necessary), and check that it lies in S t . These problems are further studied in Section 2. For R = Z, an R-module is just an abelian group; we prove two theorems that completely classify the problems P G,S and Π G,S , in the sense that for each problem we either have a polynomial time algorithm or a proof of NP-completeness.

Enriched curves have been studied over algebraically closed fields by Mainò ([Mai98]) and recently over general base schemes in [BH19]. In this paper, we study enriched curves from a logarithmic viewpoint: we give a succinct definition of the stack of rich log curves, which is an open substack of the stack of log curves, and define an enriched curve to be a curve with a minimal rich log structure on it. This logarithmic view point turns out to be a natural language for enriched structures, leading naturally to a simple