[Paper Review] Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves

Fang-Ting Tu, Yifan Yang|arXiv (Cornell University)|Dec 5, 2011

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper introduces a novel geometric approach to deriving algebraic transformations of hypergeometric functions by interpreting them as identities among automorphic forms on Shimura curves. By analyzing the modular properties and group-theoretic structure of these curves, the authors establish new degree-10 and degree-3 transformations, including a key identity linking ${}_{2}F_{1}\left(\frac{1}{20},\frac{1}{4};\frac{4}{5};\cdot\right)$ to ${}_{2}F_{1}\left(\frac{3}{10},\frac{2}{5};\frac{9}{10};z^{2}\right)$, with explicit algebraic functions R(z) and S(z).

ABSTRACT

In this paper, we will obtain new algebraic transformations of the $_2F_1$-hypergeometric functions. The main novelty in our approach is the interpretation of identities among $_2F_1$-hypergeometric functions as identities among automorphic forms on different Shimura curves.

Motivation & Objective

• To develop a geometric framework for deriving algebraic transformations of ${}_{2}F_{1}$-hypergeometric functions using automorphic forms on Shimura curves.
• To overcome limitations of classical analysis and Gröbner basis methods by leveraging the modular and group-theoretic structure of arithmetic triangle groups.
• To establish new explicit identities between hypergeometric functions with different parameters via isomorphisms of Shimura curves.
• To determine the precise algebraic functions R(z) and S(z) in transformation identities by analyzing intersection signatures of Fuchsian groups.
• To extend known results such as Goursat’s and Vidūnas’ transformations by discovering new identities of higher degree using geometric invariants.

Proposed method

• Interpret ${}_{2}F_{1}$-hypergeometric functions as automorphic forms on Shimura curves associated with quaternion algebras over number fields.
• Use the theory of arithmetic triangle groups and their subgroups to model the monodromy and singular behavior of hypergeometric differential equations.
• Analyze the signature of Fuchsian groups and their intersections via topological and group-theoretic constraints (e.g., volume, branch number, elliptic point splitting).
• Construct explicit algebraic transformations by identifying isomorphisms between Shimura curves arising from different arithmetic triangle groups.
• Apply the theory of reduced trace and norm in quaternion algebras to define modular invariants and relate them to hypergeometric parameters.
• Use subgroup diagrams and signature computations to verify that the intersection of two Fuchsian groups corresponds to a common automorphic form, implying functional identities.

Experimental results

Research questions

• RQ1 How can algebraic transformations of ${}_{2}F_{1}$-hypergeometric functions be systematically derived using geometric and modular structures?
• RQ2 What role do Shimura curves and their associated automorphic forms play in unifying known hypergeometric identities?
• RQ3 Can the algebraic functions R(z) and S(z) in transformation identities be determined via group-theoretic and topological invariants of Fuchsian groups?
• RQ4 What are the precise signatures of intersections of arithmetic triangle groups that yield new hypergeometric identities?
• RQ5 How do the degrees of transformations (e.g., degree 10) relate to the index and structure of subgroups in arithmetic triangle groups?

Key findings

• The authors derive a new degree-10 algebraic transformation: ${}_{2}F_{1}\left(\frac{5}{24},\frac{13}{24};\frac{7}{8};z\right) = R(z)\cdot{}_{2}F_{1}\left(\frac{1}{48},\frac{17}{48};\frac{7}{8};S(z)\right)$, with explicit rational functions R(z) and S(z) involving complex parameters.
• A key identity is established: ${}_{2}F_{1}\left(\frac{1}{20},\frac{1}{4};\frac{4}{5};\frac{64z(1-z-z^{2})^{5}}{(1-z^{2})(1+4z-z^{2})^{5}}\right) = (1-z^{2})^{1/20}(1+4z-z^{2})^{1/4}\cdot{}_{2}F_{1}\left(\frac{3}{10},\frac{2}{5};\frac{9}{10};z^{2}\right)$, representing a degree-3 transformation.
• The intersection of Fuchsian groups with signatures $(2,3,3,6)$ and $(2,4,6,12)$ is shown to have signature $(0;2^{3},6^{3})$, confirming the existence of a common automorphic form.
• The intersection of groups with signatures $(3,3,6)$ and $(3,4,12)$ is proven to have signature $(0;3,3,6,6)$, supporting the existence of a shared modular object.
• The intersection of groups with signatures $(2^{3},6^{3})$ and $(3,4^{2},12^{2})$ is shown to have signature $(1;2^{2},3^{2},6^{2})$, validating the consistency of the geometric construction.
• The method successfully reproduces known transformations (e.g., Kummer’s quadratic, Goursat’s degree-3) and extends them via geometric reasoning, avoiding reliance on Gröbner basis computation.