×

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 \(q(x,y,z) = (y+z)x^3 - 3x^2yz\).

MSC:

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

Software:

ecdata
PDF BibTeX XML Cite
Full Text: DOI Euclid EuDML

References:

[1] Batut C., ”User’s Guide to PARI–GP (Version 2.0) (1997)
[2] DOI: 10.1090/S0894-0347-97-00195-1 · Zbl 0872.14017
[3] Coleman R., Trans. Amer. Math. Soc. 311 pp 185– (1989)
[4] Cremona J. E., Algorithms for Modular Elliptic Curves, (1997) · Zbl 0872.14041
[5] DOI: 10.1007/978-3-540-37855-6
[6] DOI: 10.1007/BF02392133 · Zbl 0048.27503
[7] Elkies N., The Eightfold Way pp 51– (1999)
[8] DOI: 10.1090/S0025-5718-99-01123-0 · Zbl 1042.11034
[9] DOI: 10.1007/BF01677143 · JFM 11.0297.01
[10] Kraus A., Sém. de théorie des nombres de Caen (1990)
[11] Kraus A., Joumées arithmétiques de Caen (1991)
[12] DOI: 10.5802/aif.1534 · Zbl 0853.11046
[13] DOI: 10.1007/BF01444715 · Zbl 0773.14017
[14] DOI: 10.1007/978-1-4612-4752-4
[15] Ligozat G., Bull. Soc. Math. France, Supplément, mémoire 43 pp 1– (1975)
[16] DOI: 10.1007/BFb0063948
[17] DOI: 10.1007/BF01390348 · Zbl 0386.14009
[18] DOI: 10.1090/S0002-9939-1994-1212286-1
[19] Silverman J. H., The Arithmetic of Elliptic Curves (1986) · Zbl 0585.14026
[20] DOI: 10.1007/978-1-4612-0851-8
[21] Velu J., C.R. Acad. Sci. Paris, Sér. A 273 pp 238– (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.