Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians
| Autor: | Yujie Xu, Ashvin Swaminathan, Aaron Landesman, James Tao |
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| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
General theorem Mathematics - Number Theory General Mathematics Mathematics::Number Theory Group Theory (math.GR) 16. Peace & justice Galois module 11F80 11G10 11G30 11R32 Mathematics - Algebraic Geometry Mathematics::Algebraic Geometry FOS: Mathematics Torsion (algebra) Number Theory (math.NT) Representation Theory (math.RT) Abelian group Algebraic Geometry (math.AG) Mathematics - Group Theory Mathematics - Representation Theory Mathematics |
| Popis: | In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations. 25 pages |
| Databáze: | OpenAIRE |
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