𝑝-adic adelic metrics and quadratic Chabauty I
-
Amnon Besser
,
J. S. Müller
Abstract
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 ℝ-valued) on an abelian variety (and hence on pullbacks to other varieties), the local contribution at a finite prime 𝑞 can be constructed using 𝑞-analytic methods.
Funding source: Israel Science Foundation
Award Identifier / Grant number: 912/18
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: MU 4110/1-1
Funding source: Simons Foundation
Award Identifier / Grant number: 546235
Funding source: National Science Foundation
Award Identifier / Grant number: DMS 2401547
Funding statement: The first-named author was supported by grant no 912/18 from the Israel Science Foundation. The second-named author was supported by DFG grant MU 4110/1-1 and by an NWO Vidi grant. The third-named author was supported by Simons Foundation grant 546235 for the collaboration “Arithmetic Geometry, Number Theory, and Computation” and by NSF DMS 2401547. We would like to thank the NWO DIAMANT cluster for supporting a visit of the first- and third-named author to the University of Groningen.
Acknowledgements
We would like to thank Alexander Betts, Francesca Bianchi, Netan Dogra, Bas Edixhoven, Robin de Jong, Eric Katz and Klaus Künnemann for helpful discussions, and Pierre Colmez for comments on a first version of this paper. We thank the anonymous referees for particularly careful and useful reviews.
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