4. Specific functions

Problem 4.1.
Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic \[ (\sum x_i^5 + z \prod x_i =0 )/ \mathbb{Z}_5^3? \] 
Here $h_m$ is the $m$th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=1/12.$
For more information in regards to the connection between physics and number theory, see the following work of
Kathrin Bringmann and Ben Kane http://arxiv.org/pdf/1305.0112v1.pdf
Katrin Bringmann and Sameer Murthy http://arxiv.org/pdf/1208.3476v2.pdf
Katrin Bringmann and Jan Manschot http://arxiv.org/pdf/1304.7208v1.pdfProblem 4.2.
Do the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \geq 9$? 
Problem 4.3.
Does $F= \sum f_g(\tau)\lambda^{2g2}$ have transformation properties with respect to $\lambda$? 
Problem 4.4.
Is $\sum \frac{(5n)!}{(n!)^5}z^n$ related to (nonholomorphic) forms? Is it related to automorphic objects? Do other solutions to the PicardFuchs equation have modular properties?
Cite this as: AimPL: GromovWitten invariants and number theory, available at http://aimpl.org/gromwitnumthry.