Endomorphism algebras of motives attached to elliptic modular forms
Annales de l'Institut Fourier, Volume 53 (2003) no. 6, p. 1615-1676

We study the endomorphism algebra of the motive attached to a non-CM elliptic modular cusp form. We prove that this algebra has a sub-algebra isomorphic to a certain crossed product algebra X. The Tate conjecture predicts that X is the full endomorphism algebra of the motive. We also investigate the Brauer class of X. For example we show that if the nebentypus is real and p is a prime that does not divide the level, then the local behaviour of X at a place lying above p is essentially determined by the corresponding valuation of the p-th Fourier coefficient of the form.

On étudie l’algèbre des endomorphismes du motif associé à une forme modulaire parabolique sans une multiplication complexe. On démontre que cette algèbre possède une sous-algèbre isomorphe à une algèbre X de type produit croisé. La conjecture de Tate prédit que X est l’algèbre des endomorphismes du motif. On étudie également la classe de Brauer de X. Par exemple quand le nebentypus est réel et p est un nombre premier qui ne divise pas le niveau, on démontre que le comportement local de X en une place dominant p est déterminé essentiellement par la valuation correspondante du p-ième coefficient de Fourier de la forme.

DOI : https://doi.org/10.5802/aif.1989
Classification:  11G18
Keywords: endomorphism algebras, modular motives, Tate conjecture, filtered (φ,N)-modules, Newton polygons, symbols
@article{AIF_2003__53_6_1615_0,
     author = {Brown, Alexander F. and Ghate, Eknath P.},
     title = {Endomorphism algebras of motives attached to elliptic modular forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {6},
     year = {2003},
     pages = {1615-1676},
     doi = {10.5802/aif.1989},
     zbl = {1050.11062},
     mrnumber = {2038777},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_6_1615_0}
}
Brown, Alexander F.; Ghate, Eknath P. Endomorphism algebras of motives attached to elliptic modular forms. Annales de l'Institut Fourier, Volume 53 (2003) no. 6, pp. 1615-1676. doi : 10.5802/aif.1989. http://www.numdam.org/item/AIF_2003__53_6_1615_0/

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