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

$\varepsilon = \frac{t + u \sqrt{d}}{2}$

with integers t and u, it expresses in another form

$\frac{ht}{u} \pmod{p}\;$

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

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

where $m = \frac{d}{p}\;$   and  $\chi\;$  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 d=p is congruent to one mod four, then

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

where B_n is the nth Bernoulli number.

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

References

• Ankeny, N. C.; Artin, E.; Chowla, S. (1952), "The class-number of real quadratic number fields", Annals of Mathematics, Second Series 56: 479–493, doi:10.2307/1969656, MR 0049948