The 3-Part of Class Numbers of Quadratic Fields

It is proved that the 3-part of the class number of a quadratic field Q(√D) is O(|D|55/112 +ε) in general and O(|D|5/12+ε) if |D| has a divisor of size |D|5/6. These bounds follow as results of nontrivial estimates for the number of solutions to the congruence xa ≡ yb modulo q in the ranges x ⩽ X and y ⩽ Y, where a, b are nonzero integers and q is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over Q with conductor N is O(N55/112+ε) in general and O(N5/12+ε) if N has a divisor of size N5/6. These results are the first improvements to the trivial bound O(|D|1/2+ε) and the resulting bound O(N1/2+ε) for the 3-part and the number of elliptic curves, respectively.