A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two

Understand

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often.

• Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime.
• Such factorisations are expected to be modular with each quadratic factor associated to a modular form.
• If the parameter is defined over Q this modularity is assured by the proof of the Serre Conjecture.