  We compute the single-interval Renyi entropy (replica partition function) for
free fermions in 1+1d at finite temperature and finite spatial size by two
methods: (i) using the higher-genus partition function on the replica Riemann
surface, and (ii) using twist operators on the torus. We compare the two
answers for a restricted set of spin structures, leading to a non-trivial
proposed equivalence between higher-genus Siegel $\Theta$-functions and Jacobi
$\theta$-functions. We exhibit this proposal and provide substantial evidence
for it. The resulting expressions can be elegantly written in terms of Jacobi
forms. Thereafter we argue that the correct Renyi entropy for modular-invariant
free-fermion theories, such as the Ising model and the Dirac CFT, is given by
the higher-genus computation summed over all spin structures. The result
satisfies the physical checks of modular covariance, the thermal entropy
relation, and Bose-Fermi equivalence.
