Remarks on mod-ln Representations, l=3, 5 Original Research Article

Let l=3 or 5. For any integer n>1, we produce an infinite set of triples (L, E1, E2), where L is a number field with degree l3(n−1) over Q and E1 and E2 are elliptic curves over L with distinct j-invariants lying in Q, such that the following conditions hold: (1) the pairs of j-invariants {j(E1), j(E2)} are mutually disjoint, (2) the associated mod-ln representations GL=Gal(L/L)→GL2(Z/ln) are surjective, (3) for almost all primes image of L, we have ln mid aimage(E1) if and only if ln mid aimage(E2), and (4) the two representations Ei[ln](L) are not related by twisting by a continuous character GL→(Z/ln)×. No such triple satisfying (2)–(4) exists over any number field if we replace l by a prime larger than 5. The proof depends on determining the automorphisms of the group GL2(Z/ln) for l=3, 5 and analyzing ramification in a branched covering of “twisted” modular curves.