Rank gain of Jacobian varieties over finite Galois extensions
The Klein quartic is given by the homogeneous polynomial \(x^3 y + y^3 z + z^3 x = 0\), and in [\textit{N. D. Elkies}, Math. Sci. Res. Inst. Publ. 35, 51--101 (1999; Zbl 0991.11032)], various interesting properties of the Klein quartic are reviewed. The curve is isomorphic to the modular curve \(X(7)\) admitting a map down to the elliptic curve \(E :y^2=4x^3 + 21x^2 + 28x\) whose conductor is \(49\), and especially, the elliptic curve is obtained as a quotient of \(X(7)\) by a subgroup of the automorphism group of \(X(7)\). Inspired by this map on the Klein quartic, the authors of the paper under review consider a general setting, and introduce results toward the Mordell-Weil ranks of Jacobian varieties of quotient curves over certain field extensions. Let \(A\) be an abelian variety over a number field \(K\), and let \(L_i\) for \(i=1,2,\dots\) be field extensions of \(K\). The abelian variety \(A\) is said to gain rank over the extensions \(L_i\) if the Mordell-Weil rank of \(A(L_1\cdots L_n)\) is an increasing function of \(n\) where \(L_1\cdots L_n\) is the compositum of the extensions \(L_1,\dots, L_n\). Let \(p\) be a prime, and let \(\mathcal X\) be a Riemann surface of genus \(g>1\) such that there is a Galois covering \(\mathcal X \to \mathbb P_K^1\) that has degree \(p\), and is ramified only at \(0\), \(1\), and \(\infty\). The authors prove that if \(p\equiv 1\mod 6\), then there is an automorphism \(\sigma\) of \(\mathcal X\) and field extensions \(L_i/K\) of degree \(p\) such that the Jacobian of \(\mathcal X/\langle \sigma \rangle\) gains rank over the extensions \(L_i\). It is relatively easy to find any field extensions over which the Jacobian gains rank, but it becomes more difficult to find Galois extensions with rank gain, as it is essentially related to the Inverse Galois problem. The hypothesis that a curve \(\mathcal X\) admits a Galois covering of \(\mathbb P_K^1\) implies that \(K\) contains the \(p\)-th root of unity in the complex numbers, and that the field extensions \(L_i/K\) are Galois. For the Klein quartic, if \(K=\mathbb Q(e^{2\pi i/7})\), then the elliptic curve \(E\) defined above gains rank over infinitely many Galois extensions of degree \(7\) over \(K\). The Galois covering hypothesis is merely technical, and the authors introduce infinitely many examples of elliptic curves over \(\mathbb Q\) such that each member gains rank over infinitely many cyclic extensions of degree \(3\) over \(\mathbb Q\).
scientific article; zbMATH DE number 6810544
scientific article