Ankeny–Artin–Chowla congruence
# 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.

