Rational period functions, class numbers, and diophantine equations
Let \(k\in\mathbb{Z}\) and suppose \(q\) is a rational function. A modular integral \(f\) of weight \(2k\) with rational period function (RPF) \(q\) on the modular group \(\Gamma(1)\) is, by definition, a function meromorphic in \(H\), the upper half-plane, and at the cusp \(i\infty\), satisfying \[ f(z+1)=f(z), \qquad z^{-2k}f(-{1/z})=f(z)+q(z). \] In [Glasg. Math. J. 22, 185-197 (1981; Zbl 0459.10017)] the reviewer showed that the poles of an RPF of necessity lie at 0, \(\infty\) or in \(\mathbb{Q}(\sqrt{N})\), with \(N\in\mathbb{Z}^ +\). Also in [Duke Math. J. 45, 47-62 (1978; Zbl 0374.10014)] he found RPF's with poles in \(\mathbb{Q}(\sqrt{5})\). In [Ill. J. Math. 28, 383-396 (1984; Zbl 0542.10015)] \textit{L. Parson} and \textit{K. Rosen} produced examples with poles in \(\mathbb{Q}(\sqrt{3})\) and in \(\mathbb{Q}(\sqrt{21})\). The author [Ill. J. Math. 33, 495-530 (1989; Zbl 0659.10022)] was the first to construct examples with poles in \(\mathbb{Q}(\sqrt{N})\), for arbitrary \(N\in\mathbb{Z}^ +\). In the present article she presents two new constructions, this time with \(N=F_{2m}^ 2+1\), where \(F_ i\) is the \(i\)th Fibonacci number. As a corollary of her methods she proves: (i) the class number of \(\mathbb{Q}(\sqrt{F_{2m}^ 2+1})\) is \(>1\) for \(m\geq 2\); (ii) the class number of \(\mathbb{Q}(\sqrt{F_{2m}^ 2+1})\) is \(>2\) if \(m\equiv 0\pmod 3\); (iii) for \(m\geq 2\), the diophantine equation \(F_{2m+1}a^ 2-F_{2m- 1}c^ 2=1\) has no solution with \(a,c\in\mathbb{Z}\).
scientific article; zbMATH DE number 93287
scientific article