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arXiv · 2403.01500

On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements

Chan-Ho Kim
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Abstract

We explicitly construct the rank one primitive Stark (equivalently, Kolyvagin) system extending a constant multiple of Flach's zeta elements for semi-stable elliptic curves. As its arithmetic applications, we obtain the equivalence between a specific behavior of the Stark system and the minimal modularity lifting theorem, and we also discuss the cyclicity of the adjoint Selmer groups. This Stark system construction yields a more refined interpretation of the collection of Flach's zeta elements than the "geometric Euler system" approach due to Flach, Wiles, Mazur, and Weston.

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Chan-Ho Kim. 2024-03-03. On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements. https://arxiv.org/abs/2403.01500

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