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

ε

=

t

+

u

d

2

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

with integers t and u, it expresses in another form

h

t

u

(

mod

p

)

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

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

−

2

m

h

t

u

≡

∑

0

<

k

<

d

χ

(

k

)

k

⌊

k

/

p

⌋

(

mod

p

)

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

where

m

=

d

p

{\displaystyle m={\frac {d}{p}}\;}

and

χ

{\displaystyle \chi \;}

is the Dirichlet character for the quadratic field. For p = 3 there is a factor (1 + m) multiplying the LHS. Here

⌊

x

⌋

{\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

u

t

h

≡

B

(

p

−

1

)

/

2

(

mod

p

)

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

where Bn 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.