Ankeny–Artin–Chowla congruence explained

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=

	t+u\sqrt{d

}

with integers t and u, it expresses in another form

	ht
	u

\pmod{p}

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

-2{mht\overu}\equiv\sum₀{\chi(k)\overk}\lfloor{k/p}\rfloor\pmod{p}

where

	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

\lfloorx\rfloor

represents the floor function of x.

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

{u\overt}h\equivB_(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.

See also