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
:<math>\varepsilon = \frac{t + u \sqrt{d{2}</math>
with integers t and u, it expresses in another form
:<math>\frac{ht}{u} \pmod{p}\;</math>
for any prime number p > 2 that divides d. In case p > 3 it states that
:<math>-2{mht \over u} \equiv \sum_{0 < k < d} {\chi(k) \over k}\lfloor {k/p} \rfloor \pmod {p}</math>
where <math>m = \frac{d}{p}\;</math> and <math>\chi\;</math> is the Dirichlet character for the quadratic field. For p = 3 there is a factor (1 + m) multiplying the LHS. Here
:<math>\lfloor x\rfloor</math>
represents the floor function of x.
A related result is that if d=p is congruent to one mod four, then
:<math>{u \over t}h \equiv B_{(p-1)/2} \pmod{ p}</math>
where B<sub>n</sub> is the nth Bernoulli number.
There are some generalisations of these basic results, in the papers of the authors.
See also
- Herbrand–Ribet theorem, similar for ideal class groups of cyclotomic fields.