Ankeny-Artin-Chowla congruence
|
|
In number theory, the Ankeny-Artin-Chowla congruence is a result published in 1951 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
- ε = ½(t + u√d)
with integers t and u, it expresses in another form
- ht/u modulo p
for any prime number p > 2 that divides d. In case p > 3 it states that
- <math>-2{mht \over u} = \sum_{0 < k < d} {\chi(k) \over k}\lfloor {k/p} \rfloor \mod p<math>
where m = d/p, χ 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 p is congruent to one mod four, then
- <math>{u \over t}h \equiv B_{(p-1)/2} \mod p<math>
where Bn is the nth Bernoulli number.
There are some generalisations of these basic results, in the papers of the authors.