Conductor formulas and bad Euler factors for some families of CY-threefolds

We study the arithmetic of one-parameter families of Calabi–Yau threefolds with Hodge numbers h^{1,2}=h^{2,1}=1, focusing on their L-functions, in particular on the computation of bad Euler factors and the conductor. Good Euler factors can be computed using p-adic deformation methods applied to the Picard–Fuchs operators of the families. We analyse how bad Euler factors and the conductor arise from the geometry of the singular fibers, and verify this analysis by numerically checking the functional equation in examples. Special attention is given to confluence primes, where singularities collide modulo p, leading to subtle local behaviour.

Joint work in progress with Candelas, de la Ossa, van Straten.