  A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic
terms of semi-simple Frobenius structures and complex oscillating integrals is
formulated. The proof of the conjecture is given for torus-equivariant Gromov -
Witten invariants of compact K\"ahler manifolds with isolated fixed points and
for concave bundle spaces over such manifolds. Several results on genus 0
Gromov - Witten theory include: a non-linear Serre duality theorem, its
application to the genus 0 mirror conjecture, a mirror theorem for concave
bundle spaces over toric manifolds generalizing a recent result of B. Lian, K.
Liu and S.-T. Yau. We also establish a correspondence (see the extensive
footnote in section 4) between their new proof of the genus 0 mirror conjecture
for quintic 3-folds and our proof of the same conjecture given two years ago.
