Rank gain of Jacobians over number field extensions with prescribed Galois groups |
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Authors: | Bo-Hae Im Joachim König |
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Institution: | 1. Department of Mathematical Sciences, KAIST, Yuseong-gu, Daejeon, South Korea
Korea Institute for Advanced Study, Hoegiro, Seoul, South Korea;2. Department of Mathematics Education, Korea National University of Education, Cheongju, South Korea |
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Abstract: | We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has a Galois group permutation-isomorphic to a prescribed group G (in short, “G-extensions”). In particular, for alternating groups and (an infinite family of) projective linear groups G, we show that most elliptic curves over (for example) gain rank over infinitely many G-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that “many” elliptic curves gain rank over infinitely many G-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group G and certain local properties. |
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Keywords: | elliptic curve function field extensions Galois theory Jacobian variety root number |
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