On the modular curve \(X_E(7)\). (Sur la courbe modulaire \(X_E(7)\).) (French) Zbl 1050.11056

Let \(E\) be an elliptic curve defined over some field \(k\) of characteristic \(0\). For any positive integer \(n\), let \(E[n]\) denote the group of \(n\)-torsion points defined over some algebraic closure of \(k\). The action of the absolute Galois group \(G_k\) of \(k\) makes \(E[n]\) into a \(G_k\)-module, and the Weil pairing defined on \(E[n]\) is \(G_k\)-invariant.
It is known that there exists a smooth irreducible affine curve \(Y_E(n)\), unique up to \(k\)-isomorphisms, that parametrizes the isomorphism classes of pairs \((E',v)\) of elliptic curves \(E'\) and an isomorphism \(v: E[n] \to E'[n]\) compatible with the Weil pairing.
In this article, an explicit model of \(X_E(7)\) is constructed; it is a curve of genus \(3\) that becomes isomorphic to the Klein quartic over the algebraic closure of \(k\). In the special case where \(E: y^2 = (x-a)(x-b)(x-c)\) for \(a, b, c \in k\), a model of \(Y_E(7)\) is given by the smooth quartic \[ Q(x,y,z) = (c-b)q(x,y,z) + (a-c)q(y,z,x) + (b-a)q(z,x,y), \] where \(q(x,y,z) = (y+z)x^3 - 3x^2yz\).


11G05 Elliptic curves over global fields
11G18 Arithmetic aspects of modular and Shimura varieties


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