Endomorphism algebras of motives attached to elliptic modular forms
[Les algèbres d'endomorphismes des motifs associés aux formes modulaires paraboliques]
Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1615-1676.

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.

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.

DOI : 10.5802/aif.1989
Classification : 11G18
Keywords: endomorphism algebras, modular motives, Tate conjecture, filtered $(\phi ,N)$-modules, Newton polygons, symbols
Mot clés : algèbres d’endomorphismes, motifs modulaires, conjecture de Tate, $(\phi ,N)$- modules filtrés, polygones de Newton, symboles
Brown, Alexander F. 1 ; Ghate, Eknath P. 1

1 Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai 400 005 (India)
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Brown, Alexander F.; Ghate, Eknath P. Endomorphism algebras of motives attached to elliptic modular forms. Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1615-1676. doi : 10.5802/aif.1989. http://www.numdam.org/articles/10.5802/aif.1989/

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