Chevalley-Weil formula for hypersurfaces in $\mathbf{P}^n$-bundles over curves and Mordell-Weil ranks in function field towers
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866910654698356736 |
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| author | Kloosterman, Remke |
| author_facet | Kloosterman, Remke |
| contents | Let $X$ be a complex hypersurface in a $\mathbf{P}^n$-bundle over a curve $C$. Let $C'\to C$ be a Galois cover with group $G$. In this paper we describe the $\mathbf{C}[G]$-structure of $H^{p,q}(X\times_C C')$ provided that $X\times_C C'$ is either smooth or $n=3$ and $X\times_C C'$ has at most ADE singularities.% and the $\mathbf{C}[G]$-structure of the cohomology of its resolution of singularities.
As an application we obtain a geometric proof for an upper bound by Pal for the Mordell-Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface. If the Galois group of the base field acts trivially on the Galois group of the cover $C'\to C$ then we show that the bound of Pal is weaker than the bound coming from the Shioda-Tate formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1501_05184 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Chevalley-Weil formula for hypersurfaces in $\mathbf{P}^n$-bundles over curves and Mordell-Weil ranks in function field towers Kloosterman, Remke Algebraic Geometry Number Theory Let $X$ be a complex hypersurface in a $\mathbf{P}^n$-bundle over a curve $C$. Let $C'\to C$ be a Galois cover with group $G$. In this paper we describe the $\mathbf{C}[G]$-structure of $H^{p,q}(X\times_C C')$ provided that $X\times_C C'$ is either smooth or $n=3$ and $X\times_C C'$ has at most ADE singularities.% and the $\mathbf{C}[G]$-structure of the cohomology of its resolution of singularities. As an application we obtain a geometric proof for an upper bound by Pal for the Mordell-Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface. If the Galois group of the base field acts trivially on the Galois group of the cover $C'\to C$ then we show that the bound of Pal is weaker than the bound coming from the Shioda-Tate formula. |
| title | Chevalley-Weil formula for hypersurfaces in $\mathbf{P}^n$-bundles over curves and Mordell-Weil ranks in function field towers |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/1501.05184 |