Ranks of quadratic twists of Jacobians of generalized Mordell curves

Consider a two-parameter family of hyperelliptic curves Cq,b:y2=xq−bq defined over Q, and their Jacobians Jq,b where q is an odd prime and without loss of generality b is a non-zero squarefree integer. The curve Cq,b is a quadratic twist by b of Cq,1 (a generalized Mordell curve of degree q). First,...

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Bibliographic Details
Published in:Journal of algebra 2025-02, Vol.663, p.565-588
Main Author: Jędrzejak, Tomasz
Format: Article
Language:English
Subjects:
2-descent
Hyperelliptic curve
Jacobian variety
Mordell curve
Quadratic twist
Rank
Selmer group
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Summary:Consider a two-parameter family of hyperelliptic curves Cq,b:y2=xq−bq defined over Q, and their Jacobians Jq,b where q is an odd prime and without loss of generality b is a non-zero squarefree integer. The curve Cq,b is a quadratic twist by b of Cq,1 (a generalized Mordell curve of degree q). First, we obtain a few upper bounds for the ranks e.g., if the class number of Q(ζq) is odd, q≡1(mod4) and any prime divisor of 2b not equal to q is a primitive root modulo q then rankJq,b(Q)≤(q−1)/2. Then we focus on q=5 and get the best possible bound (by 1) or even the exact value of rank (0). In particular, we found infinitely many b with any number of prime factors such that rankJ5,b(Q)=0. We deduce as conclusions the complete list (or the bounds for the number) of rational points on C5,b in such cases. Finally, we found for any given q infinitely many non-isomorphic curves Cq,b such that rankJq,b(Q)≥1.
ISSN:0021-8693
DOI:10.1016/j.jalgebra.2024.08.041