Endomorphism algebras of motives attached to elliptic modular forms

Brown, Alexander F.; Ghate, Eknath P.

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.

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