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Results on Selmer groups over the $\mathrm {PGL}$
(2) extension
- Part of
-
- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 11 August 2026, pp. 1-25
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- Article
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In this article, we study Iwasawa theory for the Selmer group of an elliptic curve over a certain non-commutative p-adic Lie extension whose Galois group is isomorphic to an open subgroup of $\mathrm {PGL}_2(\mathbb {Z}_p)$
. Unlike the classes of extension usually considered in non-commutative Iwasawa theory, this extension does not contain the cyclotomic $\mathbb {Z}_p$
-extension. We give a formula for its Euler characteristic in this case. We also show that the dual Selmer group contains no nonzero pseudonull submodules, a result analogous to a theorem of Ochi and Venjakob for $\mathrm {GL}_2$
-extensions. Finally, we provide an explicit formula computing the change in Iwasawa $\mu $
-invariants of dual Selmer groups when two elliptic curves are isogenous; this in the same spirit as that of Howson’s result for the $\mathrm {GL}_2$
-extension which extends results in the commutative case by Schneider and Perrin-Riou.
