On ranks of Jacobian varieties in prime degree extensions

DOI10.4064/AA161-3-3zbMATH Open1291.11093arXiv1209.0933OpenAlexW2963678921MaRDI QIDQ2868209

Dave Mendes Da Costa

Publication date: 16 December 2013

Published in: Acta Arithmetica (Search for Journal in Brave)

Abstract: In Dokchitser (2007) it is shown that given an elliptic curve $E$ defined over a number field $K$ then there are infinitely many degree 3 extensions $L/K$ for which the rank of $E(L)$ is larger than $E(K)$. In the present paper we show that the same is true if we replace 3 by any prime number. This result follows from a more general result establishing a similar property for the Jacobian varieties associated with curves defined by an equation of the shape $g(y)=f(x)$ where $f$ and $g$ are polynomials of coprime degree.

Full work available at URL: https://arxiv.org/abs/1209.0933

zbMATH Keywords

projective curveelliptic curverankJacobian varietynumber fieldGoldfeld's conjecture

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Jacobians, Prym varieties (14H40)

Related Items (2)

Rank gain of Jacobian varieties over finite Galois extensions ⋮ Ranks in Families of Jacobian Varieties of Twisted Fermat Curves

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