Twists of Galois representations and projective automorphisms
Let \(\mathcal O\) be a commutative, complete local ring with residue field \(k\) and maximal ideal \(\lambda\). Also, let \(K\) be a finite extension of the rationals \(\mathbb Q\). Write \(G_K:=\text{Gal}(\overline{K}/K)\) and let \(\rho_1,\rho_2:G_K\rightarrow GL_n(\mathcal O)\) be surjective, continuous Galois representations, unramified outside a finite set of primes \(S\). Let \(a_i(p):=\text{Tr}(\rho_i(\text{Frob}_p))\), \(i=1,2\), \(p\not\in S\). The \(\rho_i\) are said to be \(\lambda\)-adically close near the supersingular primes (CSP, for short) if there exists an integer \(N_0>0\) such that for any integer \(w>0\), and primes \(p\) of \(K\) not in \(S\), if both \(a_i(p)\in\lambda^{N_0}\), then \(\lambda^w|a_1(p)\iff\lambda^w|a_2(p)\). One likes to know whether two such representations are `the same', i.e. are equal up to the following properties: (i) twist by a continuous character \(G_K\rightarrow{\mathcal O}^{\times}\); (ii) conjugation by a matrix in \(GL_n(\mathcal O)\); (iii) the transpose inverse automorphism of \(GL_n(\mathcal O)\); and (iv) a ring automorphism of \(\mathcal O\). With the above terminology, one can state the main result \((*)\): \(\rho_1\) and \(\rho_2\) are the same up to (i), (ii), (iii) or (iv) whenever: \((1)\) \(\mathcal O\) is a domain and \(k\not\simeq{\mathbb F}_2,{\mathbb F}_3\), or, \((2)\) \(\mathcal O\) is noetherian, \(n\geq 4\) is even and \(\text{char}(k)>2\); or \(n=2\) and \(k\not\simeq{\mathbb F}_2,{\mathbb F}_3,{\mathbb F}_5\), or \((3)\) \(\mathcal O\) is noetherian, \(n\) is odd, \(k\not\simeq{\mathbb F}_2\), and, if \(n=3\), \(k\not\simeq{\mathbb F}_3\). In the case \(n=3\) and \(k\simeq{\mathbb F}_3\) or \(n=2\) and \(\mathcal O={\mathbb Z}/\ell^N{\mathbb Z}\), \(\ell=3\) or \(5\), the result holds for the pair of representations \(G_K\rightarrow GL_n({\mathbb Z}/\ell^{N-1}{\mathbb Z})\) induced from the \(\rho_i\). The proof relies on the knowledge of the automorphisms of \(PGL_n(\mathcal O)\). This is the main theme of the paper. Building on classical results on \(\Aut(PGL_n(R))\) and \(\Aut(GL_n(R))\) for local rings, the necessary results for the complete ring \(\mathcal O\) at hand can be derived under mild conditions on \(\text{char}(k)\) and \(n\). Here cohomological tools are used. The groups \(PGL_2({\mathbb Z}/\ell^N{\mathbb Z})\), \(\ell=3\) or \(5\), and \(PGL_2\) over principal ideal domains are treated separately. With notations as before, the key result is the following lemma: Assume \((1)\) \(\mathcal O\) is an integral domain and \(k\not\simeq{\mathbb F}_2\), \(\mathbb F_3\), or \((2)\) \(n\) is even and \(k\not\simeq{\mathbb F}_2\), or \((3)\) \(n\) is odd, \(k\not\simeq{\mathbb F}_2\), and, if \(n=3\), \(k\not\simeq{\mathbb F}_3\). Then there exists an automorphism \(\phi\in \Aut(PGL_n(\mathcal O))\) such that \(\phi\circ\widetilde{\rho}_2=\widetilde{\rho}_1\), where the \(\widetilde{\rho}_i\) are the projective representations induced by the \(\rho_i\). When \(\mathcal O\) is not an integral domain, \(n\) even and \(k\simeq{\mathbb F}_2\), or \(n=3\) and \(k\simeq{\mathbb F}_3\), one obtains the same conclusion for the pair of representations \(G_K\rightarrow GL_n(\mathcal O/\lambda^N)\), where \(N\) is the largest integer such that \(\lambda^N\neq 0\) in \(\mathcal O\). Finally, a simple lifting argument leads to \((*)\). There is an interesting corollary for elliptic curves \(E_1,E_2\) defined over \(K\): (a) Let \(\rho_1,\rho_2:G_K\rightarrow GL_2({\mathbb Z}_{\ell})\) be the \(\ell\)-adic representations associated to \(E_1,E_2\). If the \(\rho_i\) are \(\ell\)-adically CSP, and either of the \(E_i\) is CM or \(\ell\) is sufficiently large, then \(E_1\) and \(E_2\) are isogenous over \(\overline{K}\); (b) Let \(\rho_1,\rho_2:G_K\rightarrow GL_2({\mathbb Z}/\ell^n{\mathbb Z})\) be the \(G_K\)-action on the \(\ell^n\)-torsion points of the \(E_i\). If the \(\rho_i\) are surjective and \(\ell\)-adically CSP, and if \(\ell>5\), then the \(\rho_i\) differ by the twist of a character \(G_K\rightarrow({\mathbb Z}/\ell^n{\mathbb Z})^{\times}\); (c) If \(\ell=3\) or \(5\), then (b) holds for the pair of representations \(G_K\rightarrow GL_2({\mathbb Z}/\ell^{n-1}{\mathbb Z})\) induced from the \(\rho_i\).
scientific article; zbMATH DE number 1271761
scientific article