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Shimura curves of discriminant 14 and 15 and associated Heun Functions

  • Harshavardhan Reddy
  • Devendra Tiwari
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Abstract

We study the Shimura curves XD*(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, He( 81, 1/2; 1/3, 1/6, 1/2, 1/2;−729/112)= ( (22 · 33 · 53)/75 )1/6.

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Cite

BibTeXRIS
BibTeX
@misc{reddy2026shimura,
  title = {{Shimura curves of discriminant 14 and 15 and associated Heun Functions}},
  author = {Harshavardhan Reddy and Devendra Tiwari},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.14495},
  eprint = {2608.14495},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14495},
}
RIS
TY  - UNPB
TI  - Shimura curves of discriminant 14 and 15 and associated Heun Functions
AU  - Harshavardhan Reddy
AU  - Devendra Tiwari
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.14495
UR  - https://arxiv.org/abs/2608.14495
AB  - 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, He( 81, 1/2; 1/3, 1/6, 1/2, 1/2;−729/112)= ( (2^2 · 3^3 · 5^3)/7^5 )^1/6.
ER  -