In Mordell *Diophantine Equations* he says:



> 
> In recent years it has been shown that there seems to be a close connection between the number of solutions of f(x,y) = 0 (mod $p^r$) and the existence of rational solutions f(x,y) = 0.
> 
> 
> 


Does anyone know what observation this is referring to? Has it been turned into a theorem?



A remark of Mordell alluding to a local/global principle for cubic Diophantine equations
As Franz says, Mordell is talking about the conjecture of Birch and Swinnerton-Dyer. But I just wanted to add that in the modern formulation of the conjecture, it is not easy to discern the original heuristic "if $E$ has lots of points modulo each prime $p$, then it should have lots of points over $\mathbb{Q}$, more precisely high rank". In fact, at the time when Birch and Swinnerton-Dyer formulated the first version of their conjecture, L-functions were largely out of fashion and nobody thought of the conjecture in this way.


If you want to see explicitly, what the modern BSD has to do with the original heuristic, you can have a look at these [hand written notes](http://www.maths.bris.ac.uk/~matyd/Parity.pdf) by Tim Dokchitser. See in particular pages 2 and 3 and top of page 5.


