Realization of some Galois representations of low degree in Mordell-Weil groups
Let \(\rho\) be a \(\mathbb{Q}\)-representation of a group \(G\). Suppose \(K/M\) is a Galois extension of number fields such that \(\text{Gal}(K/M)\cong G\), so that we may view \(\rho\) as a representation of \(\text{Gal}(K/M)\). If \(E/M\) is an elliptic curve, we will say that \(\rho\) occurs in \(E(K)\) if \(\rho\) is isomorphic to a subrepresentation of the natural representation of \(\text{Gal}(K/M)\) on \(E(K)\otimes_{\mathbb{Z}}{\mathbb{Q}}\). In the paper under review, the author shows that for certain \(G\) and \(\rho\), one may choose an arbitrary extension \(K/M\) with \(\text{Gal}(K/M)\cong G\) and be guaranteed that there exists an \(E/M\) such that \(\rho\) occurs in \(E(K)\). In particular, the author's main theorem can be stated as follows: Suppose \(G\) is a group with a subgroup \(H\) of index at most \(9\), and let \(\rho\) be the \(\mathbb{Q}\)-representation of \(G\) determined (up to isomorphism) by \[ \text{ind}_H^G\text{\textbf{1}}_H\cong\text{\textbf{1}}_G\oplus\rho, \] where \(\text{\textbf{1}}_H\) and \(\text{\textbf{1}}_G\) are the trivial representations of \(H\) and \(G\) and where \(\text{ind}_H^G\) is the induction functor from representations of \(H\) to representations of \(G\). Let \(K/M\) be an extension of number fields with \(\text{Gal}(K/M)\cong G\). Then there exists an elliptic curve \(E/M\) such that \(\rho\) occurs in \(E(K)\). The proof is by a clever direct construction. Let \(L\) be the subfield of \(K\) corresponding to the subgroup \(H\) of \(G\). One picks an element \(\xi\) of \(L\) that satisfies local conditions at two auxiliary primes \(p\) and \(q\), and uses this \(\xi\) to write down an elliptic curve \(E\) over \(M\) and a certain point \(P\) in \(E(L)\). The conditions at \(p\) and \(q\) are then used to show that the sub-\(G\)-module of \(E(K)\otimes_{\mathbb{Z}}{\mathbb{Q}}\) generated by \(P\) gives the desired representation. \textit{T. Shioda} [Mordell-Weil lattices and Galois representation I, II, III, Proc. Japan Acad., Ser. A 65, No. 7, 268-271 (1989; Zbl 0715.14015), No. 8, 296-299 (1989; Zbl 0715.14016), No. 8, 300-303 (1989; Zbl 0715.14017), and Construction of elliptic curves with high rank via the invariants of the Weyl groups, J. Math. Soc. Japan 43, No. 4, 673-719 (1991; Zbl 0751.14018)] uses a different method to find pairs \((E/M, K)\) in which certain representations -- coming from root lattices -- occur. The author notes that his method does not produce the ``largest'' of Shioda's examples, which come from \(E_6\), \(E_7\), and \(E_8\), but that his method does recover Shioda's examples of type \(A_2\) and \(D_4\). Furthermore, Shioda's examples are obtained by specializing an extension \({\mathcal K}/{\mathcal M}\) of function fields with \(\text{Gal}({\mathcal K}/{\mathcal M})\cong G\), and it is not clear at the present time whether all extensions \(K/M\) of number fields with \(\text{Gal}(K/M)\cong G\) can be obtained in this manner.
scientific article; zbMATH DE number 1019791