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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Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $$\text {PGL}_2(\mathbb {F}_q)$$ PGL 2 ( F q ) or $$\text {PSL}_2(\mathbb {F}_q)$$ PSL 2 ( F q ) . We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.

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