In their early papers
entitled 'Notes on Elliptic Curves', Birch and Swinnerton-Dyer formulated
a conjecture about the convergence of certain Euler products. This conjecture
was a precursor that led to the more refined and celebrated conjecture
relating ranks of elliptic curves to orders of zeros of L-functions. We
will show that the 'original' Birch and Swinnerton-Dyer conjecture lies
far far beyond the Riemann hypothesis and that it is likely to be true,
but unlikely to proved in this millennium.
This is joint work with Ram Murty.
For more details on the thematic year, see Program