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SIMULTANEOUS BOUNDING OF
$4$-RANKS IN NARROW CLASS GROUPS OF QUADRATIC FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 113 / Issue 3 / June 2026
- Published online by Cambridge University Press:
- 29 January 2026, pp. 440-446
- Print publication:
- June 2026
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- Article
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This paper studies the
$4$-ranks of narrow class groups in certain families of quadratic fields. We prove that for any positive integer n, there exists an integer 
$s_\lambda (n)$ depending on n and the sign 
$\lambda $ of the fundamental discriminant D, such that for any choice of 
$s_\lambda (n)$ integers 
$t_1, \ldots , t_{s_\lambda (n)}$, there are infinitely many D for which the narrow class group of 
$\mathbb {Q}(\sqrt {D + t_i})$ has 
$4$-rank bounded by n for all i. This result extends previous work on 
$3$-ranks to the case of 
$4$-ranks.
On elliptic curves with p-isogenies over quadratic fields
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 75 / Issue 3 / June 2023
- Published online by Cambridge University Press:
- 07 June 2022, pp. 945-964
- Print publication:
- June 2023
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Let K be a number field. For which primes p does there exist an elliptic curve
$E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.
