The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds

A banana manifold is a Calabi-Yau threefold fibered by Abelian surfaces whose singular fibers contain banana configurations: three rational curves meeting each other in two points. A nano-manifold is a Calabi-Yau threefold X with very small Hodge numbers: h1,1(X)+h2,1(X)≤ 6. We construct four rigid banana nano-manifolds X̃N, N∈ \5,6,8,9 \, each with Hodge numbers given by (h1,1,h2,1)=(4,0). We compute the Donaldson-Thomas partition function for banana curve classes and show that the associated genus g Gromov-Witten potential is a genus 2 meromorphic Siegel modular form of weight $2g-2$ for a certain discrete subgroup P*N ⊂ Sp₄(R). We also compute the weight 4 modular form whose pth Fourier coefficient is given by the trace of the action of Frobenius on H³ₑₜ(X̃N ,Qₗ) for almost all prime p. We observe that it is the unique weight 4 cusp form on Γ₀(N).