LECTURE 5. SOME APPLICATIONS OF SUBCONVEXI'IY 283When q is squarefree at least, one can check that this conjecture follows from
ScP for the £-functions (S.14) in the level aspect.
Analogously to remark 5.8, this conjecture would have the pleasing conse-
quence that the projections of the zeros of f by n q become equidistributed on
X 0 (1), relative to μ p. A suggestive corollary is that the multiplicity of any such
zero is o(q) as q __, +oo. this improves over the Riemann/ Roch theorem, which
bounds the multiplicity by O(q n plq(l + 1/ p)). When f = f E corresponds to an
elliptic curve, the zeros are the ramification points of the modular parametrization
E, and have multiplicity negligible by comparison with the conductor. Such corol-
laries would certainly be meaningful in connection with the abc-conjecture, as very
little is known about the ramification divisor of a strong Weil curve (it is not even
clear that the ramification cannot be concentrated on one point).