On ranks of Jacobian varieties in prime degree extensions (Q2868209)
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scientific article; zbMATH DE number 6238627
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On ranks of Jacobian varieties in prime degree extensions |
scientific article; zbMATH DE number 6238627 |
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On ranks of Jacobian varieties in prime degree extensions (English)
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16 December 2013
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Jacobian variety
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elliptic curve
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number field
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rank
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Goldfeld's conjecture
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projective curve
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The author generalizes a result of \textit{T. Dokchitser} [Acta Arith. 126, No. 4, 357--360 (2007; Zbl 1125.11032)], and prove that for an elliptic curve \(E\) which is defined over a number field \(K\), there exists infinitely many degree \(p\) extensions \(L/K\) where \(p\) is a prime number (the case \(p=3\) was studied by Dokchitser), for which the rank of \(E(L)\) is greater than \(E(K)\). This follows from a more general similar property for the Jacobian varieties associated with curves defined by an equation of the form \(f(y)= g(x)\) with \(f\), \(g\) being polynomials of coprime degree.NEWLINENEWLINE The author's approach is inspired by the above-mentioned paper of T. Dokchitser.
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