Arakelov-Green's functions for dynamical systems on projective varieties
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916199585021952 |
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| author | Looper, Nicole R. |
| author_facet | Looper, Nicole R. |
| contents | We introduce functions associated to polarized dynamical systems that generalize averages of the dynamical Arakelov-Green's functions for rational functions due to Baker and Rumely. For a polarized dynamical system $X\to X$ over a product formula field, we prove an Elkies-style lower bound for these functions evaluated on the adelic points of $X$. As an application, we prove a Lehmer-type lower bound on the canonical height of a non-torsion point $P$ on an abelian variety $A/K$, where $K$ is a product formula field having perfect residue fields at its completions (for instance, $K$ may be a number field or the function field of a curve over $\mathbb{C}$ or $\mathbb{F}_p$). For $A$ of dimension $g$, the lower bound has the form \[\widehat{h}(P)\ge\frac{C}{D^{2g+3}(\log D)^{2g}},\] where $C=C(A,K,\widehat{h})>0$, $D=[K(P):K]\ge2$, and $P\in A(\bar{K})\setminus A(\bar{K})_{\text{tors}}$ is not contained in a torsion translate of an abelian subvariety of $A$ having everywhere potential good reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arakelov-Green's functions for dynamical systems on projective varieties Looper, Nicole R. Number Theory Algebraic Geometry Dynamical Systems 11G10, 37P30 (Primary) 14G40, 14K15, 37P55 (Secondary) We introduce functions associated to polarized dynamical systems that generalize averages of the dynamical Arakelov-Green's functions for rational functions due to Baker and Rumely. For a polarized dynamical system $X\to X$ over a product formula field, we prove an Elkies-style lower bound for these functions evaluated on the adelic points of $X$. As an application, we prove a Lehmer-type lower bound on the canonical height of a non-torsion point $P$ on an abelian variety $A/K$, where $K$ is a product formula field having perfect residue fields at its completions (for instance, $K$ may be a number field or the function field of a curve over $\mathbb{C}$ or $\mathbb{F}_p$). For $A$ of dimension $g$, the lower bound has the form \[\widehat{h}(P)\ge\frac{C}{D^{2g+3}(\log D)^{2g}},\] where $C=C(A,K,\widehat{h})>0$, $D=[K(P):K]\ge2$, and $P\in A(\bar{K})\setminus A(\bar{K})_{\text{tors}}$ is not contained in a torsion translate of an abelian subvariety of $A$ having everywhere potential good reduction. |
| title | Arakelov-Green's functions for dynamical systems on projective varieties |
| topic | Number Theory Algebraic Geometry Dynamical Systems 11G10, 37P30 (Primary) 14G40, 14K15, 37P55 (Secondary) |
| url | https://arxiv.org/abs/2404.06981 |