Abstract
In this article, we show that for any nonisotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density- subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension , there are infinitely many abelian varieties over with adelic Galois representation having image equal to all of .
Citation
Aaron Landesman. Ashvin A. Swaminathan. James Tao. Yujie Xu. "Surjectivity of Galois representations in rational families of abelian varieties." Algebra Number Theory 13 (5) 995 - 1038, 2019. https://doi.org/10.2140/ant.2019.13.995
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