Ankeny-Artin-Chowla congruence

In number theory, the Ankeny-Artin-Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla. It concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is

ε = ½(t + u√d)

with integers t and u, it expresses in another form

ht/u modulo p

for any prime number p > 2 that divides d. In case p > 3 it states that

$-2{mht \over u} = \sum_{0 < k < d} {\chi(k) \over k}\lfloor {k/p} \rfloor \mod p$

where m = d/p, χ is the Dirichlet character for the quadratic field. For p = 3 there is a factor (1 + m) multiplying the LHS. Here

$\lfloor x\rfloor$

represents the floor function of x.

A related result is that if p is congruent to one mod four, then

${u \over t}h \equiv B_{(p-1)/2} \mod p$

where B_n is the nth Bernoulli number.

There are some generalisations of these basic results, in the papers of the authors.

[edit] Reference

N. C. Ankeny, E. Artin, S.Chowla, The class-number of real quadratic number fields, Annals of Math. 56 (1953), 479-492