Rational period functions, class numbers, and diophantine equations

AbstractWe construct several rational period functions for modular integrals with weight 2k on the modular group Γ(1). It is also possible to represent the above rational period functions in terms of representatives of reduced indefinite binary quadratic forms in the narrow equivalence class. As a corollary, we are able to show that the class number of the real quadratic fields Q(√F2m2 + 1) for m = 2, 3, … is bigger than 1, where Fi is the ith Fibonacci number. Moreover, we are able to show that the class number of the real quadratic field Q(√F2m2 + 1), for m = 0 (mod 3), m ≥ 2, is bigger than 2