Real Quadratic Fields and the Ankeny–Artin–Chowla Conjecture

Ankeny-Artin-Chowla type congruences modulo p 3

“…to rewrite (16). In the resulting congruence use (2) and (3) to show that the terms that are not multiples of p cancel each other.…”

“…For the history of the above congruence and conjecture see [16]. All of this is quite important but we would like to look for explicit congruences of the same type that are valid for a different class of fields and modulo higher powers of p. In this direction, S. Jakubec, using algebraic techniques in [3][4][5][6][7][8][9], has established congruences of Ankeny-Artin-Chowla type modulo p and p 2 for totally real cyclic fields K of degree l and prime conductor p. Perhaps his technique is best summarized in [14: Section 1], the notation of which we are going to follow (reader is encouraged to consult [14] if needed).…”