  We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau
hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti
Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a
quintic threefold. We combine constructions from a series of previous papers
with the classical localization theorem to relate the reduced genus 1
invariants of a CY-hypersurface to previously computed integrals on moduli
spaces of stable genus 0 maps into projective space. The resulting, rather
unwieldy, expressions for a genus 1 equivariant generating function simplify
drastically, using a regularity property of a genus 0 equivariant generating
function in half of the cases. Finally, by disregarding terms that cannot
effect the non-equivariant part of the former, we relate the answer to an
explicit hypergeometric series in a simple way. The approach described in this
paper is systematic. It is directly applicable to computing reduced genus 1
GW-invariants of other complete intersections and should apply to higher-genus
localization computations.
