The other part of the Birch-Swinnerton-Dyer conjecture (which we will not state here) is a precise formula for the first non-vanishing derivative of L(E,s) at s = 1; it is an example of a class number formula. The earliest examples of such formulas are related to the failure of unique factorization in number fields and are due to Lejeune Dirichlet. For example, the Dirichlet class number formula for the field Q(√-5) is
2⁄2 · 3⁄2 · 5⁄5 · 7⁄6 · 11⁄12 · 13⁄14 &hellip = &pi⁄ √5where the numerators run over prime numbers p and the denominators are p - 1 if p ≡ 1,3,7,9 mod 20 and p + 1 if p ≡ 11,13,17,19 mod 20 (and simply p if p is 2 or 5).
One side of a class number formula is the value of an L-function at an integer point. In the case of Q(√-5) the L-function is
L(Q (√-5), s) := &prodp ( 1 - (-5⁄p)p-s )-1with (-5/p) the Legendre symbol of -5 modulo p. The left side of the Dirichlet class number formula is simply L(Q (√-5),1).
The right side consists of three factors. One is a regulator, which is usually a transcendental number of some sort; in this case it is &pi. The second is a relatively uninteresting nuisance factor; in this case, it is actually 1⁄ 2√5. The third factor, 2 in this case, is the class number. In general, the class number should be related to some interesting arithmetic fact. In this case, it is the size of the ideal class group of Z[√-5]; the fact that it is different from 1 implies that Z[√-5] does not have unique factorization, as is also evident from the famous counterexample
2 · 3 = (1 + √-5) · (1 - √-5).Previous: Elliptic curves