Ankeny–Artin–Chowla congruence

Short description: Concerns the class number of a real quadratic field of discriminant > 0

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]\displaystyle{ \varepsilon = \frac{t + u \sqrt{d}}{2} }[/math]

with integers t and u, it expresses in another form

[math]\displaystyle{ \frac{ht}{u} \pmod{p}\; }[/math]

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

[math]\displaystyle{ -2{mht \over u} \equiv \sum_{0 \lt k \lt d} {\chi(k) \over k}\lfloor {k/p} \rfloor \pmod {p} }[/math]

where [math]\displaystyle{ m = \frac{d}{p}\; }[/math]   and  [math]\displaystyle{ \chi\; }[/math]  is the Dirichlet character for the quadratic field. For p = 3 there is a factor (1 + m) multiplying the LHS. Here

[math]\displaystyle{ \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]\displaystyle{ {u \over t}h \equiv B_{(p-1)/2} \pmod{ p} }[/math]

where B_n is the nth Bernoulli number.

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

References

• "The class-number of real quadratic number fields", Annals of Mathematics, Second Series 56 (3): 479–493, 1952, doi:10.2307/1969656, http://repository.ias.ac.in/8797/1/395.pdf