Compatible systems and companion completion

Pointwise automorphy at one coefficient place and existence of companions at other coefficient places are different statements. They meet only after a common algebraic Frobenius spine has been fixed.

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Finite forests alone give compatible classes. To obtain actual continuous representations, we use the following explicit coefficient and admissibility data. They also delimit the meaning of completion in the next theorem.

Fix a topological group \(\Gamma\), a finite matrix index set \(D\), and a set \(\Lambda\) of coefficient indices. For each \(\lambda\), let \({\mathcal O}_\lambda\) be a commutative ring, complete and separated for a finitely generated ideal \(I_\lambda\), with finite residue ring \({\mathcal O}_\lambda/I_\lambda\). Give \({\mathcal O}_\lambda\) its ideal adic topology and every quotient its discrete topology. Let

be supplied ring homomorphisms, where \(L\) and \(K_\lambda\) are fields, \(K_\lambda\) has a declared topology, and \(i_\lambda\) is continuous.

For every positive integer \(n\), let \(X_{\lambda,n}\) be a finite nonempty set of conjugacy classes of actual continuous representations

Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that \(X_{\lambda,1}\) consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.

Choose elements \(\gamma_v\in\Gamma\) on a sample set \(\Omega\), with declared good subsets \(\Omega_\lambda\subseteq\Omega\). In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each \(v\in\Omega_\lambda\), supply an integral polynomial \(P_{\lambda,v}\in{\mathcal O}_\lambda[X]\) such that

Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \(X_{\lambda,n}\) has characteristic polynomial \(P_{\lambda,v}\bmod I_\lambda^n\) at \(\gamma_v\). Also supply a subgroup \(H_\lambda\leq\Gamma\) on which all these finite representations are trivial. This is an inertia condition when \(H_\lambda\) is the specified inertia subgroup.

Finally, put

For each declared local condition index \(u\), supply an ideal \(J_{\lambda,u}\subseteq B_\lambda\). Evaluation at a representation sends \(x_{g,a,b}\) to its \((a,b)\) matrix entry at \(g\). Assume that evaluation at every admissible finite representative kills \(J_{\lambda,u}\), with polynomial coefficients reduced modulo \(I_\lambda^n\). For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitrary arithmetic condition is identified with such an ideal without a separate argument.

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The quotient conclusion is a proved effectivity statement for the declared polynomial relations. For local conditions presented by continuous maps from represented local rings, Corollary 3.2 also accounts for the closed ideal generated by their defining ideals. Neither statement constructs a deformation ring for a condition that has not been shown representable. Hodge conditions, Weil–Deligne conditions, integral lifts of arbitrary Frobenius polynomials, and the finite occupancy hypotheses still require their own arithmetic inputs. The theorem retains a residual conjugacy class, not a prescribed residual matrix frame, and asserts no uniqueness up to integral conjugacy.

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