Realization of Some Galois Representations of Low Degree in Mordell-weil Groups

For example, suppose that G ∼= S9 and H ∼= S8, where Sn denotes the symmetric group on n letters. Then ρ is one of the two absolutely irreducible representations of G of dimension 8, and according to our theorem there exists an elliptic curve E over M such that ρ occurs in Q⊗E(K). The other absolutely irreducible representation of G of dimension 8 is ρ ⊗ , where is the “sign” character of G, and it too can be realized in a Mordell-Weil group: in fact a straightforward argument shows that if ρ occurs in Q ⊗ E(K) then ρ ⊗ occurs in Q ⊗ E (K), where E denotes the quadratic twist of E by . The case G ∼= S9, H ∼= S8 just mentioned is actually the maximal instance of the theorem, in two respects: first, for any choice of G and H the dimension of ρ will be 6 8, and second, if we assume without loss of generality that K is the normal closure of L over M then G is always isomorphic to a subgroup of S9. It follows in particular that G is not isomorphic to one of the Weyl groups W (E6), W (E7), or W (E8), because these groups do not have embeddings in S9. Thus we do not recover the “biggest” examples of Shioda ([8], [9], [10]), whose work on Mordell-Weil lattices of elliptic surfaces yields examples of type E6, E7, and E8 by specialization. Here for a root system X the phrase “example of type X” means a Galois extension of number fields K/M together with an elliptic curve E over M and an identification of G = Gal(K/M) with W (X) such that the representation of G on Q⊗E(K) has a subrepresentation isomorphic to the representation of W (X) on the rational span of X. In addition to examples of type E6, E7, and E8, Shioda’s theory also produces examples of type A2 and D4, and the latter cases fall within the framework of