  The classical Lefschetz fixed point theorem states that the number of fixed
points, counted with multiplicity $\pm 1$, of a smooth map $f$ from a manifold
$M$ to itself can be calculated as the alternating sum $\sum (-1)^k \textrm{ tr
} f^*|_{H^k(M)}$ of the trace of the induced homomorphism in cohomology. In
1964, at a conference in Woods Hole, Shimura conjectured a Lefschetz fixed
point theorem for a holomorphic map, which Atiyah and Bott proved and
generalized into a fixed point theorem for elliptic complexes. However, in
Shimura's recollection, he had conjectured more than the holomorphic Lefschetz
fixed point theorem. He said he had made a conjecture for a holomorphic
correspondence, but he could not remember the statement. This paper is an
exploration of Shimura's forgotten conjecture, first for a smooth
correspondence, then for a holomorphic correspondence in the form of two
conjectures and finally in the form of an open problem involving an extension
to holomorphic vector bundles over two varieties and the calculation of the
trace of a Hecke correspondence.
