AI 中文总结
本文研究判别式14和15的Shimura曲线，将其四元数模形式环生成元与Heun函数关联，分析阿贝尔曲面族的Picard-Fuchs方程解，明确Heun函数的有理例外集及对应代数值。
AI 中文摘要
判别式D=14、15的Shimura曲线由算术四边形Fuchs群(2,2,2,q)的子群一致化，其中q=4、6。我们将该Shimura曲面上四元数模形式环的生成元与该四边形群的显式Heun函数关联起来。我们还讨论了相关阿贝尔曲面族的Picard-Fuchs方程的解如何为X^D(1)/W_D上的模形式，其中W_D是完整的Atkin-Lehner对合群。这使我们能完全描述相关Heun函数的有理例外集，以及该函数在这些点上取得的代数值，例如He(81, 1/2; 1/3, 1/6, 1/2, 1/2; -729/112) = (2²·3³·5³ / 7⁵)^(1/6)。
英文摘要
The Shimura curve of discriminant $D$ for $D=14, 15$ is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group $(2, 2, 2, q)$, where $q=4, 6$. We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms on $X^{D}(1) / W_{D}$, where $W_D$ is the full group of Atkin-Lehner involutions. This leads us to completely describe the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function on these points, for example $${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}}$$.
