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# Signed-Selmer Groups over the $\mathbb{Z}_p^2$-extension of an Imaginary Quadratic Field

Published:2013-11-14
Printed: Aug 2014
• Byoung Du Kim,
Victoria University of Wellington
 Format: LaTeX MathJax PDF

## Abstract

Let $E$ be an elliptic curve over $\mathbb Q$ which has good supersingular reduction at $p\gt 3$. We construct what we call the $\pm/\pm$-Selmer groups of $E$ over the $\mathbb Z_p^2$-extension of an imaginary quadratic field $K$ when the prime $p$ splits completely over $K/\mathbb Q$, and prove they enjoy a property analogous to Mazur's control theorem. Furthermore, we propose a conjectural connection between the $\pm/\pm$-Selmer groups and Loeffler's two-variable $\pm/\pm$-$p$-adic $L$-functions of elliptic curves.
 Keywords: elliptic curves, Iwasawa theory
 MSC Classifications: 11Gxx - Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx]

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