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

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

with integers t and u, it expresses in another form

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

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

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

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

${\displaystyle \lfloor x\rfloor }$

represents the floor function of x.

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

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

where B_n is the nth Bernoulli number.

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

See also

• Herbrand–Ribet theorem, similar for ideal class groups of cyclotomic fields.

References

• Ankeny, N. C.; Artin, E.; Chowla, S. (1952), "The class-number of real quadratic number fields" (PDF), Annals of Mathematics, Second Series, 56 (3): 479–493, doi:10.2307/1969656, JSTOR 1969656, MR 0049948