Uniform boundedness of small points on abelian varieties over function fields
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866910069648523264 |
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| author | Looper, Nicole Yap, Jit Wu |
| author_facet | Looper, Nicole Yap, Jit Wu |
| contents | Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that any $K$-rational torsion point $x$ of $A$ has order uniformly bounded in terms of $g$ and the gonality of $B$. We also prove a uniform lower bound on the Néron-Tate height $\widehat{h}_{A,L}(x)$ in terms of the stable Faltings height $h_{\mathrm{Fal}}(A)$ for any $K$-rational point $x$ whose forward orbit is Zariski dense, proving the Lang-Silverman conjecture over function fields of characteristic $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23396 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform boundedness of small points on abelian varieties over function fields Looper, Nicole Yap, Jit Wu Number Theory Algebraic Geometry Dynamical Systems 11G10, 14G40, 14K15, 32P05, 37P30 Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that any $K$-rational torsion point $x$ of $A$ has order uniformly bounded in terms of $g$ and the gonality of $B$. We also prove a uniform lower bound on the Néron-Tate height $\widehat{h}_{A,L}(x)$ in terms of the stable Faltings height $h_{\mathrm{Fal}}(A)$ for any $K$-rational point $x$ whose forward orbit is Zariski dense, proving the Lang-Silverman conjecture over function fields of characteristic $0$. |
| title | Uniform boundedness of small points on abelian varieties over function fields |
| topic | Number Theory Algebraic Geometry Dynamical Systems 11G10, 14G40, 14K15, 32P05, 37P30 |
| url | https://arxiv.org/abs/2603.23396 |