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?
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 by Tim Dokchitser. See in particular pages 2 and 3 and top of page 5.
For the sake of the history, as told to me by Cassels: Birch and Swinnerton-Dyer initially were looking at the "rate of divergence" of the infinite products that are formally what contribute to L(1). In other words there are a few steps to take from Mordell's version:
(1) The difference between the p-adic and mod p counting is really not serious, given Hensel's lemma, away from a finte number of primes.
(2) The first heuristic is that Hasse's theorem on elliptic curves over finite fields tells you that N(p), the number of points mod p, is p + error where the error is of the order of the square root of the main term.
(3) The finite products of the N(p)/p were calculated for given curves by Birch and Swinnerton-Dyer, and plotted on logarithmic graph paper. This didn't have any particular right to work, but did.
(4) At a key moment Birch and Swinnerton-Dyer, who were now confident that their plots had predictive value for the rank, but were encountering scepticism, were told (I believe by Cassels) that it would be sensible to look at the L-function. One has to realise that the analytic continuation to the point in question at which the whole heuristic apparatus could even work was quite unclear, outside the case of complex multiplication known from Deuring.
In short, this is all much more like the way physics works.
