Hecke operator and Pellian equation conjecture. II (Q1192411)

[Part I (not reviewed).] Let \(p\equiv 1\pmod 4\) be a prime and \(\varepsilon=(t+u\sqrt{p})/2\) be the least solution of the Pell equation \(x^ 2-py^ 2=-4\). The conjecture due to N. C. Ankeny, E. Artin and S. Chowla says that \(p\nmid u\). In this paper the author finds a relation between this conjecture and the Hecke operator on the Hirzebruch sum. For a real quadratic irrational number \(\beta\) which has a development of the simple continued fraction \([\widehat a_ 0,\dots,\widehat a_ s,\overline {a_ 1,\dots,a_ k}]\) with the basic period \(\overline{a_ 1,\dots,a_ k}\), define the Hirzebruch sum \(\psi(\beta)=\sum_{j=1}^ k (-1)^{j+s} a_ j\) if \(k\) is even and \(\psi(\beta)=0\) if \(k\) is odd. Let \(T_ N\) be the Hecke operator, i.e. \(T_ N(f)(\alpha)={\displaystyle {\sum_{{mn=N, m>0} \atop {\omega\pmod m}}}} f({{n\alpha+\omega} \over m})\) for a positive integer \(N\) and a function \(f\) of \(\alpha\). The author shows that \((T_{p^ k}\psi)(2\sqrt{p})= \lambda_ k \psi(2\sqrt{p})\). The value of \(\lambda_ k\) is also given. Furthermore, define the Dirichlet series \(J_ p(s)= \sum_{k=0}^ \infty \lambda_ k p^{-ks}\) (\(\text{Re}(s)>3\)). The author shows that \(J_ p(s)\) is a rational function of \(p^{-s}\) and points out that \(J_ p(s)=(1-p^{-s})^{-1} (1-p^{2-s})^{-1}\), which is the zeta function of \(\mathbb{P}'(\mathbb{F}_{p^ 2})\), if and only if \(p \nmid u\).