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
y2 = x3 + ax + bfor some rational numbers a,b. Such a curve is called an elliptic curve. If E is an elliptic curve, then E(Q) may be finite or infinite, depending on E. Surprisingly, however, if E(Q) is finite, then by a theorem of Barry Mazur the only possibilities for its size are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14 and 16.
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 Fp for various primes p and somehow piece this information together to recover E(Q). That is, for a prime number p let E(Fp) denote the set of solutions (x,y) with x,y ∈ Fp of
y2 ≡ x3 + ax + b (mod p)(together with a single point at infinity). A theorem of Hasse states that E(Fp) usually has around p + 1 elements, so we let
ap(E) := p + 1 - #E(Fp)denote the deviation from this expected value. (The numbers ap(E) are sometimes called the Fourier coefficients of E. For certain elliptic curves E they are closely related to class field theory. In this direction, Ravi Ramakrishna suggested to me that it might be interesting to determine how often ap(E) is a cube modulo p; see Power residues of Fourier coefficients of modular forms for some first results on this problem.)
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
L(E,s) := &prodp (1 - app-s + p1-2s)-1where the infinite product is over all prime numbers p. (We are actually being a little sloppy here; our Euler factors are not quite right for a few primes. This will not cause any immediate problems, fortunately.) This product converges if Re(s) > 3⁄ 2, and by the Taniyama-Shimura conjecture (now a theorem of Andrew Wiles, Richard Taylor, Fred Diamond, Brian Conrad and Christophe Breuil) it is known that L(E,s) can be analytically continued to a function defined for all complex numbers s.
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
"L(E,1) = &prodp p⁄ #E(Fp)".(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 paper Partial Euler products on the critical line by Keith Conrad.) In order for L(E,1) to vanish we would thus need #E(Fp) to usually be larger than p. As we said before, we would expect #E(Fp) to on average be larger than p if E(Q) is infinite. The Birch-Swinnerton-Dyer conjecture is a precise version of this intuition. Previous: Arithmetic geometry