  The mathematical physicists Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed,
in a seminal article from '94, a conjecture extending genus zero mirror
symmetry to higher genera. With a view towards a refined formulation of the
Grothendieck-Riemann-Roch theorem, we offer a mathematical description of the
BCOV conjecture at genus one. As an application of the arithmetic Riemann-Roch
theorem of Gillet-Soul\'e and of our previous results on the BCOV invariant, we
establish this conjecture for Calabi-Yau hypersurfaces in projective spaces.
Our contribution takes place on the $B$-side, and together with the work of
Zinger on the $A$-side, it provides the first complete examples of the mirror
symmetry program in higher dimensions. The case of quintic threefolds was
studied by Fang-Lu-Yoshikawa. Our approach also lends itself to arithmetic
considerations of the BCOV invariant, and we study a Chowla-Selberg type
theorem expressing it in terms of special $\Gamma$ values for certain
Calabi-Yau manifolds with complex multiplication.
