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## 4. Specific functions

1. #### 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.pdf

#### Problem 4.2.

Do the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \geq 9$?
• #### Problem 4.3.

Does $F= \sum f_g(\tau)\lambda^{2g-2}$ have transformation properties with respect to $\lambda$?
• #### Problem 4.4.

Is $\sum \frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects? Do other solutions to the Picard-Fuchs equation have modular properties?

Cite this as: AimPL: Gromov-Witten invariants and number theory, available at http://aimpl.org/gromwitnumthry.