Mazur ’ s question on mod 11 representations of elliptic curves ∗

The main idea of the proof of this Theorem is to study the geometry (and arithmetic) of the modular diagonal quotient surfaces ZN,1 (as introduced in [9]) in the special case N = 11. Now the algebraic surface Z = ZN,1 has a natural model as a variety over Q (cf. §3), and an open subvariety of this turns out to be the coarse moduli space of the moduli functor ZN,1 which classifies isomorphism classes of triplets (E1, E2, ψ), where ψ : E1[N ] → E2[N ] is a Gal(Q̄/Q)-isomorphism which preserves the Weil pairings. Thus, via this modular interpretation (cf. §4), the above Theorem is essentially a consequence of the following result (cf. Theorem 19): ∗This research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).