Remarks on mod-\(l^n\) representations, \(l=3,5\)
Let \(\ell\) be an odd prime and \(n > 1\) an integer. For an elliptic curve \(E/K\) over a number field \(K\), denote by \(a_{\mathfrak p}(E)\) the trace on the Tate module \(T_\ell(E)\) of an arithmetic Frobenius element at \(\mathfrak p\). Suppose that the associated mod-\(\ell^n\) Galois representation \(\rho_{E, \ell^n} : G_K \to \mathop{GL}_2({\mathbb Z}/\ell^n {\mathbb Z})\) is surjective. In an earlier paper by \textit{S. Wong} [J. Number Theory 74, 1-18 (1999; Zbl 0931.11016)], it was proved that for \(\ell > 5\), \(\rho_{E, \ell^n}\) is determined up to twisting by a continuous character by the set of primes \(\mathfrak p\) outside a finite set such that \(\ell^n\) divides \(a_{\mathfrak p}\). In the paper under review, the authors complement this result by showing that it is no longer true for \(\ell = 3\) and \(\ell = 5\). More precisely, there is a number field \(L\) of absolute degree \(\ell^{3(n-1)}\) such that there are infinitely many pairs \((E_1, E_2)\) of elliptic curves over \(L\), all with distinct \(j\)-invariants, such that (1) \(\rho_{E_1, \ell^n}\) and \(\rho_{E_2, \ell^n}\) are both surjective, (2) for almost all primes \(\mathfrak p\) of \(L\), \(\ell^n\) divides \(a_{\mathfrak p}(E_1)\) if and only if it divides \(a_{\mathfrak p}(E_2)\), (3) \(\rho_{E_1, \ell^n}\) and \(\rho_{E_2, \ell^n}\) are not related by twisting by a continuous character \(G_L \to ({\mathbb Z}/\ell^n {\mathbb Z})^\times\). For the proof, one starts with a curve \(E/K\) and its representation \(\rho = \rho_{E, \ell^n}\). The presence of a non-trivial outer automorphism of \(\mathop{PGL}_2({\mathbb Z}/\ell^n {\mathbb Z})\) allows to construct another representation \(\rho'\) such that \(\rho\) and \(\rho'\) have the required properties. The final step is to realize \(\rho'\) as a representation coming from an elliptic curve. This is done by analyzing the corresponding modular curve \(X(\rho')\). At this point, it is necessary to enlarge the base field in order to obtain rational points on this curve.
scientific article; zbMATH DE number 1375090