Endomorphism algebras of motives attached to elliptic modular forms

Brown, Alexander F.; Ghate, Eknath P.

doi:10.5802/aif.1989

Endomorphism algebras of motives attached to elliptic modular forms
[Les algèbres d'endomorphismes des motifs associés aux formes modulaires paraboliques]

Brown, Alexander F.¹ ; Ghate, Eknath P.¹

¹ Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai 400 005 (India)

Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1615-1676.

Résumé
Abstract

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.

MR Zbl

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

Affiliations des auteurs :

Brown, Alexander F. ¹ ; Ghate, Eknath P. ¹

¹ Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai 400 005 (India)

@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.1989/}
}

TY  - JOUR
AU  - Brown, Alexander F.
AU  - Ghate, Eknath P.
TI  - Endomorphism algebras of motives attached to elliptic modular forms
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 1615
EP  - 1676
VL  - 53
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1989/
DO  - 10.5802/aif.1989
LA  - en
ID  - AIF_2003__53_6_1615_0
ER  -

%0 Journal Article
%A Brown, Alexander F.
%A Ghate, Eknath P.
%T Endomorphism algebras of motives attached to elliptic modular forms
%J Annales de l'Institut Fourier
%D 2003
%P 1615-1676
%V 53
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1989/
%R 10.5802/aif.1989
%G en
%F 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, Tome 53 (2003) no. 6, pp. 1615-1676. doi : 10.5802/aif.1989. https://aif.centre-mersenne.org/articles/10.5802/aif.1989/

Bibliographie
Cité par

[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 | DOI | MR | Zbl

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

[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 $ℓ$ -adiques, Séminaire Bourbaki, 1968/1969 (Lecture Notes in Math.), Volume 179, exp. 355 (1971), pp. 139-172 | Numdam | Zbl

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

[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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

[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

[Rib92] K. Ribet Abelian varieties over $ℚ$ 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 | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[Ste00] W. Stein The first newform such that $ℚ (a_{n}) \neq ℚ (a_{1}, a_{2}, ...)$ for all $n$ (2000) (Preprint)

[Vol01] M. Volkov Les représentations $ℓ$ -adiques associées aux courbes elliptiques sur $ℚ_{p}$ , J. reine angew. Math., Volume 535 (2001), pp. 65-101 | DOI | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT