On the \(f\)-invariant of products (Q505350)

\textit{J. F. Adams} [Topology 5, 21--71 (1966; Zbl 0145.19902)] considered the \(d\)-invariant and the notion of an \(e\)-invariant in his work on the \(J\)-homomorphism. The work of \textit{M. F. Atiyah} et al. [Math. Proc. Camb. Philos. Soc. 77, 43--69 (1975; Zbl 0297.58008)] on index theory on manifolds with boundary led to an analytic formulation of the \(e\)-invariant [\textit{M. F. Atiyah} et al., ibid. 78, 405--432 (1975; Zbl 0314.58016)]. \textit{Ch. Deninger} and \textit{W. Singhof} [Invent. Math. 78, 101--112 (1984; Zbl 0558.55010)] used the \(e\)-invariant to exhibit an infinite family of nilmanifolds representing the generator of Im\(J\) in dimensions \(8k+3\) \((8k+7)\). \textit{G. Laures} [Topology 38, No. 2, 387--425 (1999; Zbl 0924.55004); Trans. Am. Math. Soc. 352, No. 12, 5667--5688 (2000; Zbl 0954.55008)] introduced the \(f\)-invariant which is a follow-up to the \(e\)-invariant and takes values in the divided congruences between modular forms. In this paper, by using some elementary number theory, the author shows that if \(x_{4k-1}\) is a generator of Im\(J\) in dimension \(4k-1\) and if \(\mu_{8k+1}\) is a representative of the \(\mu\)-family, then, for the level \(N=3\), \(f(x_3^2) \equiv {1 \over 2} ( {{E_1^2 -1} \over {12} })^2\); \(f(x_7^2) \equiv {1 \over 2} ( {{E_4 -1} \over {240} })^2\); \(f(\mu_{8k+1} \mu_{8k'+1}) \equiv {1 \over 2} {{E_1 -1} \over {2} }\); \(f(\mu_{8k+1} x_{8k'-1}) \equiv {1 \over 2} {{E_4 -1} \over {240} }\); and \(f(\mu_{8k+1} x_{8k'+3}) \equiv 0\); and that for any level \(N > 1\), \(f(x_{4k-1} x_{4k'-1}) \equiv 0\), unless \(k =k'\) equals one or two.