Remarks on mod-I-n representations, I=3, 5

Let l = 3 or 5. For any integer n > 1, we produce an infinite set of triple s ( L, E-1, E-2), where L is a number field with degree l(3(n-1)) over Q an d E-1 and E-2 are elliptic curves over L with distinct j-invariants lying i n Q, such that the following conditions hold: (1) the pairs of j-invariants {j(E-1), j(E-2)} are mutually disjoint, (2) the associated mod-l(n) repres entations G(L) = Gal((L) over bar/L) --> GL(2)(Z/l(n)) are surjective: (3) for almost all primes rho of L, we have l(n) \ a(rho)(E-1) if and only if l (n) \ a(rho)(E-2), and (4) the two representations E-i[l(n)]((L) over bar) are not related by twisting by a continuous character G(L) --> (Z/l(n))(x) . No such triple satisfying (2)-(4) exists over any number field if we repl ace I by a prime larger than 5. The pl oof depends on determining the autom orphisms of the group GL(2)(Z/l(n)) for l= 3, 5 and analyzing ramification in a branched covering of "twisted" modular curves. (C) 1999 Academic Press .