From this point of view the most interesting curves are thus those of genus one. If a curve of genus one has at least one rational point, then it can be given as the zero locus of a cubic equation

for some rational numbersy^{2}=x^{3}+ax+b

In general it is difficult to determine directly for a given elliptic
curve whether it has finitely many or infinitely many rational points.
Arithmetic geometers do have at least one trick, known as the
*Hasse principle*, which is often useful:
rather than trying to find the points of *E* over the infinite field
**Q**, instead try to find them over the finite field
**F**_{p} for various primes *p* and somehow piece this
information together to recover *E*(**Q**).
That is, for a prime
number *p* let *E*(**F**_{p}) denote the set of
solutions (*x*,*y*) with *x*,*y* ∈
**F**_{p} of

(together with a single point at infinity). A theorem of Hasse states thaty^{2}≡x^{3}+ax+b(modp)

denote the deviation from this expected value. (The numbersa(_{p}E) :=p+ 1 - #E(F_{p})

Since one can reduce points of *E*(**Q**) modulo primes *p*,
if *E*(**Q**) is infinite, then one might expect
*E* to usually have more points than expected modulo *p*.
The
*conjecture of Birch and Swinnerton-Dyer*
makes this prediction
explicit. Define the *L-function* of *E* as the function of
the complex variable *s* defined by

where the infinite product is over all prime numbersL(E,s) := &prod_{p}(1 -a+_{p}p^{-s}p^{1-2s})^{-1}

The first part of the Birch-Swinnerton-Dyer conjecture states that
*E*(**Q**) is infinite if and only if *L*(*E*,1) = 0.
To relate this to the intuition we mentioned previously, note that
formally plugging *s* = 1 into the product for
*L*(*E*,*s*) yields

"(We have put this equality in quotes because, as was originally observed by Dorian Goldfeld, it is not actually true; see for example the recent paperL(E,1) = &prod_{p}^{p}⁄_{#E(Fp)}".