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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].

11R23 Iwasawa theory
11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms