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 : https://doi.org/10.5802/aif.1989
Classification : 11G18
Mots clés : algèbres d’endomorphismes, motifs modulaires, conjecture de Tate, (φ,N)- modules filtrés, polygones de Newton, symboles
@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},
     pages = {1615--1676},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {6},
     year = {2003},
     doi = {10.5802/aif.1989},
     zbl = {1050.11062},
     mrnumber = {2038777},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1989/}
}
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/

[AL78] A.O.L. Atkin; Wen Ch'ing Winnie Li Twists of newforms and pseudo-eigenvalues of W-operators, Invent. Math., Volume 48 (1978) no. 3, pp. 221-243 | Article | MR 508986 | Zbl 0369.10016

[BR93] D. Blasius; J. Rogawski. Motives for Hilbert modular forms, Invent. Math., Volume 114 (1993), pp. 55-87 | Article | MR 1235020 | Zbl 0829.11028

[Bre01] C. Breuil Lectures on p-adic Hodge theory, deformations and local Langlands, Advanced Course Lecture Notes, Volume 20

[Del69] P. Deligne Formes modulaires et représentations -adiques, Séminaire Bourbaki, 1968/1969 (Lecture Notes in Math.), Volume 179, exp. 355 (1971), pp. 139-172 | Numdam | Zbl 0206.49901

[Dem72] M. Demazure Lectures on p-divisible groups, Lecture Notes in Math., 302, Springer-Verlag, 1972 | MR 344261 | Zbl 0247.14010

[DR73] P. Deligne; M. Rapoport Les schémas de modules de courbes elliptique, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Math.), Volume Vol. 349 (1973), pp. 143-316 | Zbl 0281.14010

[Fal83] G. Faltings Endlichkeitssätze für abelsche Varietäten über Zahlkörpen, Invent. Math., Volume 73 (1983), pp. 349-366 | Article | MR 718935 | Zbl 0588.14026

[FM83] J.-M. Fontaine; B. Mazur Geometric Galois representations, Ser. Number Theory, 1, International Press, 1995 | MR 1363495 | Zbl 0839.14011

[Hid00] H. Hida Modular Forms and Galois Cohomology, Cambridge University Press, Cambridge, 2000 | MR 1779182 | Zbl 0952.11014

[Jan92] U. Jannsen Motives, numerical equivalence, and semi-simplicity, Invent. Math., Volume 107 (1992) no. 3, pp. 447-452 | Article | MR 1150598 | Zbl 0762.14003

[Mom81] F. Momose On the -adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., Volume 28 (1981), pp. 89-109 | MR 617867 | Zbl 0482.10023

[Quer98] J. Quer La classe de Brauer de l'algèbre d'endomorphismes d'une variété abélienne modulaire, C. R. Acad. Sci. Paris, Sér. I Math., Volume 327 (1998) no. 3, pp. 227-230 | Article | MR 1650241 | Zbl 0936.14032

[Rib80] K. Ribet Twists of modular forms and endomorphisms of abelian varieties, Math. Ann., Volume 253 (1980) no. 1, pp. 43-62 | Article | MR 594532 | Zbl 0421.14008

[Rib81] K. Ribet Endomorphism algebras of abelian varieties attached to newforms of weight 2, Seminar on Number Theory, Paris 1979-1980 (Progr. Math.), Volume 12 (1981), pp. 263-276 | Zbl 0467.14006

[Rib92] K. Ribet Abelian varieties over and modular forms, Proc. KAIST Math. Workshop (1992), pp. 53-79

[Sch90] A. Scholl Motives for modular forms, Invent. Math., Volume 100 (1990), pp. 419-430 | Article | MR 1047142 | Zbl 0760.14002

[Ser81] J.-P. Serre Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci., Volume 54 (1981), pp. 323-401 | Numdam | MR 644559 | Zbl 0496.12011

[Shi71] G. Shimura On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J., Volume 43 (1971), pp. 199-208 | MR 296050 | Zbl 0225.14015

[Shi73] G. Shimura On the factors of the Jacobian variety of a modular function field, J. Math. Soc. Japan, Volume 25 (1973), pp. 523-544 | Article | MR 318162 | Zbl 0266.14017

[Ste00] W. Stein The first newform such that (a n )(a 1 ,a 2 ,...) for all n (2000) (Preprint)

[Vol01] M. Volkov Les représentations -adiques associées aux courbes elliptiques sur p , J. reine angew. Math., Volume 535 (2001), pp. 65-101 | Article | MR 1837096 | Zbl 1024.11038