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p-adic adelic metrics and Quadratic Chabauty I

We give a new construction of $p$-adic heights on varieties over number fields using $p$-adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of $p$-adic adelic metrics on line bundles. In particular, we describe a construction of canonical $p$-adic heights on abelian varieties and we show that we recover the canonical Mazur--Tate height and, for Jacobians, the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle. We show that our construction allows us to reprove, without using $p$-adic Hodge theory or arithmetic fundamental groups, several results due to Balakrishnan and Dogra. Our method also extends to primes $p$ of bad reduction. One consequence of our work is that for any canonical height ($p$-adic or $\mathbb{R}$-valued) on an abelian variety (and hence on pull-backs to other varieties), the local contribution at a finite prime $q$ can be constructed using $q$-analytic methods.