×

Some computations with Hecke rings and deformation rings. With an appendix by Amod Agashe and William Stein. (English) Zbl 1116.11310

Summary: In the proof by Wiles, completed by Taylor-Wiles, of the fact that all semistable elliptic curves over \(\mathbb Q\) are modular, certain deformation rings play an important role. In this note, we explicitly compute these rings for the elliptic curve \(Y^2+XY=X^3-X^2-X-3\) of conductor 142.

MSC:

11F80 Galois representations
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)

Software:

ecdata
PDF BibTeX XML Cite
Full Text: DOI Euclid EuDML

References:

[1] Cornell G., Modular forms and Fermat’s Last Theorem (1997) · Zbl 0878.11004
[2] Birch B. J., Modular Functions of One Variable IV (1975)
[3] Cremona J., Algorithms for modular elliptic curves (1992) · Zbl 0758.14042
[4] Darmon H., Seminar on Fermat’s Last Theorem 1993–1994 pp 135– (1995)
[5] De Shalit E., Modular Forms and Fermat’s Last Theorem (1997)
[6] De Smit B., Modular Forms and Fermat’s Last Theorem (1997)
[7] Diamond F., Elliptic Curves, Modular Forms and Fermat’s Last Theorem pp 22– (1995)
[8] Diamond F., Modular Forms and Fermat’s Last Theorem (1997)
[9] Edixhoven S. J., Modular Forms and Fermat’s Last Theorem (1997)
[10] Gelbart S., Modular Forms and Fermat’s Last Theorem (1997)
[11] Mazur B., Modular Forms and Fermat’s Last Theorem. (1997)
[12] Rio A., Tesi Doctoral, in: Representacions de Galois octaédriques (1995)
[13] Serre J.-P., Duke Math. Journal 54 pp 179– (1987) · Zbl 0641.10026
[14] Serre J.-P., Invent. Math. 15 pp 259– (1972) · Zbl 0235.14012
[15] Stein W. A., HECKE package, Magma V2.7 or higher (2000)
[16] Sturm J., Number theory (New York, 1984–1985) pp 275– (1987)
[17] Taylor R., Ann. Math. 141 pp 553– (1995) · Zbl 0823.11030
[18] Tilouine J., Modular Forms and Fermat’s Last Theorem (1997)
[19] Wiles A., Ann. Math. 141 pp 443– (1995) · Zbl 0823.11029
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.