Math @ Duke

Publications [#302450] of Lillian B. Pierce
Papers Published
 Pierce, LB, The 3part of class numbers of quadratic fields,
Journal of the London Mathematical Society, vol. 71 no. 3
(2005),
pp. 579598, ISSN 00246107 [doi]
(last updated on 2018/08/17)
Abstract: It is proved that the 3part of the class number of a quadratic field ℚ(√D) is O(D55/112+ε) in general and O(D 5/12+ε) if D has a divisor of size D5/6. 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 squarefree 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 3part and the number of elliptic curves, respectively. © 2005 London Mathematical Society.


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