Descent properties for an abelian variety with extended Galois representation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915778159181824 |
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| author | Felder, Ludovic |
| author_facet | Felder, Ludovic |
| contents | Let $K$ be a field, $L$ a finite Galois extension of $K$, and $X$ an abelian variety defined over $L$. If $X$ is isogenous over $L$ to an abelian variety defined over $K$, then the $\ell$-adic Galois representations associated to $X$ extend to representations $\barρ_{\ell,X}:\mathrm{Gal}(\bar{L}/K)\to\mathrm{Aut}(V_\ell X)$ for every prime $\ell$. This paper aims to show that the converse is true for abelian varieties of Type I, with some supplementary conditions needed on the endomorphisms of $X$, when $L$ is either a number field or a function field of prime characteristic different from $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05834 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Descent properties for an abelian variety with extended Galois representation Felder, Ludovic Number Theory Algebraic Geometry 11R34, 11R80, 11G10, 14K15 Let $K$ be a field, $L$ a finite Galois extension of $K$, and $X$ an abelian variety defined over $L$. If $X$ is isogenous over $L$ to an abelian variety defined over $K$, then the $\ell$-adic Galois representations associated to $X$ extend to representations $\barρ_{\ell,X}:\mathrm{Gal}(\bar{L}/K)\to\mathrm{Aut}(V_\ell X)$ for every prime $\ell$. This paper aims to show that the converse is true for abelian varieties of Type I, with some supplementary conditions needed on the endomorphisms of $X$, when $L$ is either a number field or a function field of prime characteristic different from $2$. |
| title | Descent properties for an abelian variety with extended Galois representation |
| topic | Number Theory Algebraic Geometry 11R34, 11R80, 11G10, 14K15 |
| url | https://arxiv.org/abs/2602.05834 |