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On Selmer groups of adjoint modular Galois representations. (English) Zbl 0924.11090
David, Sinnou (ed.), Number theory. Séminaire de Théorie des Nombres de Paris 1993–94. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 235, 89-132 (1996).
Fix an odd prime $$p$$. Let $$\mathbb{I}$$ denote the universal ordinary Hecke algebra of level $$N$$, $$(N,p)=1$$. Let $$\varphi$$ be a modular Galois representation into $$GL_2(\mathbb{I})$$, and let $$\nu$$ be the universal character unramified outside $$p$$ deforming the trivial character of $$\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})$$. Consider the Selmer group $$\text{Sel(Ad} (\varphi)\otimes \nu^{-1})/ \overline{\mathbb{Q}}$$ of Greenberg for the adjoint representation $$\text{Ad} (\varphi)$$ on the trace zero subspace $$V(\text{Ad} (\varphi))$$ of $$M_2(\mathbb{I})$$. The author proves, under a suitable assumption, a control theorem of the deformation rings (Theorem 2.2). He then deduces the control theorem for the Selmer group $$\text{Sel(Ad} (\varphi)\otimes \nu^{-1})/ \overline{\mathbb{Q}}$$ (Theorem 3.2), and its co-torsionness (Theorem 3.3).
For the proof he considers the universal ordinary ring $$R_F$$ of $$\varphi$$ restricted to $$\text{Gal} (\overline{\mathbb{Q}}/F)$$, and identifies $$\text{Sel(Ad} (\varphi))/F$$ with the module of 1-differentials of $$R_F$$.
The article also includes corrections (pp. 129-132) to the author’s article [Lond. Math. Soc. Lect. Note Ser. 215, 139-166 (1995; Zbl 0838.11031)].
For the entire collection see [Zbl 0898.00026].

##### MSC:
 11R23 Iwasawa theory 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms