In Swinnerton-Dyer's charming paper "An application of computing to classfield theory", in Cassels-Frohlich, he discusses the genesis of the Birch/Swinnerton-Dyer conjecture and numerical tests of it for the curves $y^2=x^3-dx$. At the end of the paper, he speculates on higher-dimensional analogues of the conjecture, linking Chow groups to L-functions, which in hindsight were essentially correct (Beilinson-Bloch made them precise). Regarding these conjectures, he says that Bombieri and himself had found some evidence for them in the special case of cubic threefolds and the intersection of two quadric hypersurfaces. I know that in the case of cubic threefolds $X$, codimension-two cycles can be related to zero-cycles in the Albanese $A_X$ of the Fano surface of lines in the cubic surface, which (I assume) reduces the conjecture in this case to BSD for $A_X$. But what can be done for the intersection of two quadrics? Does anyone know what he was talking about here? Is this in print in more detail somewhere?

In Swinnerton-Dyer's charming paper "An application of computing to classfield theory", in Cassels-Frohlich, he discusses the genesis of the Birch/Swinnerton-Dyer conjecture and numerical tests of it for the curves $y^2=x^3-dx$. At the end of the paper, he speculates on higher-dimensional analogues of the conjecture, linking Chow groups to L-functions, which in hindsight were essentially correct (Beilinson-Bloch made them precise). Regarding these conjectures, he says that Bombieri and himself had found some evidence for them in the special case of cubic threefolds and the intersection of two quadric hypersurfaces. I know that in the case of cubic threefolds $X$, codimension-two cycles can be related to zero-cycles in the Albanese $A_X$ of the Fano surface of lines in $X$, which (I assume) reduces the conjecture in this case to BSD for $A_X$. But what can be done for the intersection of two quadrics? Does anyone know what he was talking about here? Is this in print in more detail somewhere?

In Swinnerton-Dyer's charming paper "An application of computing to classfield theory", in Cassels-Frohlich, he discusses the genesis of the Birch/Swinnerton-Dyer conjecture and numerical tests of it for the curves $y^2=x^3-dx$. At the end of the paper, he speculatives on higher-dimensional analogues of the conjecture, linking Chow groups to L-functions, which in hindsight were essentially correct (Beilinson-Bloch made them precise). Regarding these conjectures, he says that Bombieri and himself had found some evidence for them in the special case of cubic threefolds and the intersection of two quadric hypersurfaces. I know that in the case of cubic threefolds $X$, codimension-two cycles can be related to zero-cycles in the Albanese $A_X$ of the Fano variety of lines in the cubic surface, which (I assume) reduces the conjecture in this case to BSD for $A_X$. But what can be done for the intersection of two quadrics? Does anyone know what he was talking about here? Is this in print in more detail somewhere?

In Swinnerton-Dyer's charming paper "An application of computing to classfield theory", in Cassels-Frohlich, he discusses the genesis of the Birch/Swinnerton-Dyer conjecture and numerical tests of it for the curves $y^2=x^3-dx$. At the end of the paper, he speculates on higher-dimensional analogues of the conjecture, linking Chow groups to L-functions, which in hindsight were essentially correct (Beilinson-Bloch made them precise). Regarding these conjectures, he says that Bombieri and himself had found some evidence for them in the special case of cubic threefolds and the intersection of two quadric hypersurfaces. I know that in the case of cubic threefolds $X$, codimension-two cycles can be related to zero-cycles in the Albanese $A_X$ of the Fano surface of lines in the cubic surface, which (I assume) reduces the conjecture in this case to BSD for $A_X$. But what can be done for the intersection of two quadrics? Does anyone know what he was talking about here? Is this in print in more detail somewhere?