On the modular curve \(X_{\text{ndép}}(11)\). (Sur la courbe modulaire \(X_{\text{ndép}}(11)\).) (French) Zbl 0922.11047

Halberstadt, Emmanuel

Exp. Math. 7, No. 2, 163-174 (1998).

If \(p \geq 5\) is a prime number, the modular curves \(X_{\text{dép}}(p) = X_{\text{split}}(p)\) and \(X_{\text{ndép}}(p) = X_{\text{nonsplit}}(p)\) classify elliptic curves \(E\) such that the Galois action on \(E[p]\) factors through the normaliser of a split (resp. non-split) Cartan subgroup of \(\text{GL}_2({\mathbb F}_p)\). The paper under review is mainly concerned with the curve \(X_{\text{nonsplit}}(11)\). It was known before [G. Ligozat, Modular functions of one variable, Lect. Notes Math. 601, 149-237 (1977; Zbl 0357.14006)] that this curve is an elliptic curve isomorphic over \(\mathbb Q\) to the curve \[ {\mathcal E} : y^2 + y = x^3 - x^2 - 7x + 10. \] The author makes this isomorphism explicit and proceeds to find explicitly the rational function on \({\mathcal E}\) giving the \(j\)-invariant. This in turn leads to two explicit examples of pairs \((E, E')\) of non-isogenous elliptic curves over \(\mathbb Q\) with symplectically isomorphic mod 11 Galois representations.
In an appendix, it is shown that the curve \(X_{\text{split}}(37)\) has no extra rational points (i.e., other than cusps and CM points). F. Momose [Compos. Math. 52, 115-137 (1984; Zbl 0574.14023)] had shown that there is at most one such point. The proof given here makes use of these results, together with knowledge on \(X_0(37)\) taken from B. Mazur and P. Swinnerton-Dyer [Invent. Math. 25, 1-61 (1974; Zbl 0281.14016)].

Reviewer: Michael Stoll (Düsseldorf)

MSC:

11G05	Elliptic curves over global fields
11G18	Arithmetic aspects of modular and Shimura varieties
14H52	Elliptic curves

Keywords:

elliptic curves over number fields; mod \(p\) Galois representations; \(j\)-invariant

Citations:

Zbl 0357.14006; Zbl 0574.14023; Zbl 0281.14016

Software:

ecdata

References:

[1]	Cassou-Noguès P., Elliptic functions and rings of integers (1987) · Zbl 0621.12012
[2]	Cremona J. E., Algorithms for modular elliptic curves (1992) · Zbl 0758.14042
[3]	Deligne P., dans Modular functions of one variable (Antwerp, 1972) pp 143– (1973) · doi:10.1007/978-3-540-37855-6_4
[4]	Halberstadt E., C. R. Acad. Sci. Pans Sér. I Math. 322 (4) pp 313– (1996)
[5]	Kubert D., Math. Ann. 218 (2) pp 175– (1975) · doi:10.1007/BF01370818
[6]	Lang S., Elliptic functions (1987) · doi:10.1007/978-1-4612-4752-4
[7]	Ligozat G., Modular functions of one variable, V (Bonn, 1976) pp 149– (1977) · doi:10.1007/BFb0063948
[8]	Mazur B., Invent. Math. 44 (2) pp 129– (1978) · Zbl 0386.14009 · doi:10.1007/BF01390348
[9]	Mazur B., Invent. Math. 25 pp 1– (1974) · Zbl 0281.14016 · doi:10.1007/BF01389997
[10]	Momose F., Compositio Math. 52 (1) pp 115– (1984)
[11]	Serre J.-P., Invent. Math. 15 (4) pp 259– (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086
[12]	Serre J.-P., Lectures on the MordellWeil theorem (1989)
[13]	Shimura G., Introduction to the arithmetic theory of automorphic functions (1971) · Zbl 0221.10029
[14]	Silverman J. H., The arithmetic of elliptic curves (1986) · Zbl 0585.14026 · doi:10.1007/978-1-4757-1920-8
[15]	Velu J., C. R. Acad. Sci. Paris Sér. A-B 273 pp A238– (1971)