Zeta functions and modularity of Calabi-Yau manifolds

We study the Dwork operator U(λ) on a four dimensional subspace of the Dwork cohomology in the large structure limit of a one parameter family of quintic threefolds. The Dwork cohomology becomes singular in this limit, but we recover it by introducing formal logarithmic states which remain well defined at λ = 0. In a suitable basis, U(0) is known to have a single nonzero off-diagonal component at the large structure point, whose value is a rational multiple of ζp(3). We compute the value of this off-diagonal component via a direct computation in our "logarithmic Dwork cohomology" at λ =0$, and obtain an expansion for U(λ) around λ=0. This expansion provides an efficient means to compute a quartic factor R(T) of the local zeta function, not just for the quintic but for a large number of one parameter families of Calabi-Yau manifolds. We exploit the miraculous fact that U(λ) becomes a rational function mod p4 to avoid evaluating the series for U(λ) directly. This is vital to the performance of the computation for large primes. The behaviour of the zeta function at singular values of the parameter is particularly interesting. The Weil conjectures ensure rationality, but say nothing about the form of the zeta function when the variety is singular. What generally happens is that one or more eigenvalues of U(λ) go to zero and the degree of R(T) falls. Typically a quadratic factor occurs in R(T) with which a modular form can be associated. For a number of one parameter families, we evaluate R(T) at the singular points, for 7 ⤠p ⤠97, and seek to identify the associated modular form. We reproduce various known results with higher certainty by checking up to p=97. We identify modular forms in many new cases, including Hilbert modular forms of parallel weight 4 at irrational conifold singularities.