A reciprocity law for prime geodesics
This paper concerns a closed geodesic \(p\) on the surface \(\text{SL}(2,\mathbb Z)\setminus \mathcal H\) and the corresponding primitive hyperbolic conjugacy class \(\{\beta\}\) in the modular group \(\Gamma =\text{SL}(2,\mathbb Z)\) of all \(2\times 2\) integer matrices of determinant one. Here \(\mathcal H\) is the upper half plane and \(\Gamma\) acts by fractional linear transformation. Suppose that \(\Gamma(q)\) is the congruence subgroup of \(\Gamma\) consisting of all matrices congruent to the identity matrix modulo \(q\). The paper under review shows that the way \(p\) splits when lifted to the surface \(\Gamma(q)\setminus \mathcal H\) depends only on the length \(\ell\) of \(p\). Here \(q\) is an odd prime. In fact, the splitting of \(p\) on the covering surface depends only on the residue class mod \(q^ 2\) of the integer \(t=2 \cosh (\ell /2).\) The second result in this paper gives a reciprocity law saying that the behavior of \(p\) when lifted to \(\Gamma_ 0(q)\setminus \mathcal H\) is equivalent to the factorization of \(q\) in the ring of integers of the number field \(\mathbb Q(\varepsilon)\), where \(\varepsilon\) is an eigenvalue of an element of \(\text{Im}\,s<T.\) The proof essentially depends on the nonvanishing of \(L(s)\) for \(s>11/2\), \(s\in \mathbb R\); on \textit{A. Good}'s mean-value theorem [Mathematika 29, 278--295 (1982; Zbl 0503.10020)]; on an estimate for \(\sum_{m\leq n}\tau (m)\), due to \textit{A. Perelli} [Mathematika 31, 150--158 (1984; Zbl 0549.10033)]; and on calculations by \textit{R. C. Spira} [Math. Comput. 27, 379--385 (1973; Zbl 0283.10022)] that ensure existence of a zero \(6+i\gamma_ 0\) of odd multiplicity with \(9<\gamma_ 0<10\) for \(L(s)\). The authors have announced their result in C. R. Math. Acad. Sci., Soc. R. Canada 10, No. 2, 95--99 (1988; Zbl 0652.10030)].
scientific article; zbMATH DE number 4065083