On an Archimedean analogue of Tate's conjecture. (Q1869817)

The aim of this article is to present the following conjecture. Conjecture: Suppose that \(X\) and \(Y\) are complete algebraic curves defined over a number field \(K\). Suppose that the compact Riemann surfaces associated to \(X\) and \(Y\), via an embedding of \(K\) into \(\mathbb C\), are isospectral with respect to the Kähler metric of constant curvature \(-1\). Then the Hasse-Weil zeta-functions of \(X\) and \(Y\) are the same over a finite extension \(L\) of \(K\), and hence the Weil restriction of scalars from \(L\) to \(\mathbb Q\) of the corresponding Jacobians are isogenous. The authors explain the relation of this conjecture to Sunada's construction of isospectral manifolds [\textit{T. Sunada}, Ann. Math. (2) 121, 169--186 (1985; Zbl 0585.58047)]. In particular, they show that the above conjecture holds for the isospectral Riemann surfaces constructed by Vignéras und Sunada. The conjecture can be considered as an Archimedean analogue of Tate's conjecture.