## The 3-part of class numbers of quadratic fields

Publication
, Journal Article

Pierce, LB

Published in: Journal of the London Mathematical Society

2005

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. © 2005 London Mathematical Society.

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## Published In

Journal of the London Mathematical Society

## DOI

## Publication Date

2005

## Volume

71

## Issue

3

## Start / End Page

579 / 598

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics

### Citation

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Pierce, L. B. (2005). The 3-part of class numbers of quadratic fields.

*Journal of the London Mathematical Society*,*71*(3), 579–598. https://doi.org/10.1112/S002461070500637XPierce, L. B. “The 3-part of class numbers of quadratic fields.”

*Journal of the London Mathematical Society*71, no. 3 (2005): 579–98. https://doi.org/10.1112/S002461070500637X.Pierce LB. The 3-part of class numbers of quadratic fields. Journal of the London Mathematical Society. 2005;71(3):579–98.

Pierce, L. B. “The 3-part of class numbers of quadratic fields.”

*Journal of the London Mathematical Society*, vol. 71, no. 3, 2005, pp. 579–98.*Scival*, doi:10.1112/S002461070500637X.Pierce LB. The 3-part of class numbers of quadratic fields. Journal of the London Mathematical Society. 2005;71(3):579–598.

## Published In

Journal of the London Mathematical Society

## DOI

## Publication Date

2005

## Volume

71

## Issue

3

## Start / End Page

579 / 598

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics