Congruence of Ankeny-Artin-Chowla

In Theory of the numbers, the congruence of Ankeny-Artin-Chowla is a result published in 1951 by N.C. Ankeny, Emil Artin and S. Chowla. It relates to the many classes H of a quadratic Corps real of Discriminant D > 0. If the fundamental Unité of the body is

$\ epsilon = \ frac {1} {2} (T + U \ sqrt {D}) \,$

with T and U whole, it expresses in another form

$\ frac {HT} {U} \ equiv \ pmod {p} \,$

for all Prime number p > 2 which divides D . In the case p > 3, it establishes:

$-2 {mht \ over U} \ equiv \ sum_ {0 < K < D} {\ chi (K) \ over K} \ lfloor {k/p} \ rfloor \ MOD p$

where $m = \ frac {D} {p} \,$ , $\ chi \,$ is the Caractère of Dirichlet for the quadratic body. For p = 3, there exists a factor (1 + m ) multiplying the left side of the equation. Here,

$\ lfloor X \ rfloor$

represent the function whole Partie X .

A result connected is the following: if $p \ equiv 1 \ MOD 4 \,$ , then

${U \ over T} H \ equiv B_ {\ frac {(p-1)}{2}} \ MOD p$

Where B_n is nth the Nombre of Bernoulli.

There exist certain generalizations of these results of bases in the articles of the authors.

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