Year: 2026
Type: preprint
Abstract: We study the Shimura curves $X_D^*(1)$ of discriminant $D$ for $D=14, 15$ relating specific weight $4$ modular forms on these Shimura curves to the associated Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions coming from modular forms on $X^*_{D}(1)$. We determine the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function at these points. For example, $${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}} $$.
Source: arXiv (Cornell University)
Authors: Reddy, Harshavardhan, Tiwari, Devendra
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203597425