Endomorphism algebras of motives attached to elliptic modular forms

Alexander F. Brown[1]; Eknath P. Ghate[1]

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

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 6, page 1615-1676
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Brown, Alexander F., and Ghate, Eknath P.. "Endomorphism algebras of motives attached to elliptic modular forms." Annales de l’institut Fourier 53.6 (2003): 1615-1676. <http://eudml.org/doc/116082>.

@article{Brown2003,
abstract = {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.},
affiliation = {Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai 400 005 (India); Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai 400 005 (India)},
author = {Brown, Alexander F., Ghate, Eknath P.},
journal = {Annales de l’institut Fourier},
keywords = {endomorphism algebras; modular motives; Tate conjecture; filtered $(\phi ,N)$-modules; Newton polygons; symbols; filtered ; N)},
language = {eng},
number = {6},
pages = {1615-1676},
publisher = {Association des Annales de l'Institut Fourier},
title = {Endomorphism algebras of motives attached to elliptic modular forms},
url = {http://eudml.org/doc/116082},
volume = {53},
year = {2003},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 6
SP - 1615
EP - 1676
AB - 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.
LA - eng
KW - endomorphism algebras; modular motives; Tate conjecture; filtered $(\phi ,N)$-modules; Newton polygons; symbols; filtered ; N)
UR - http://eudml.org/doc/116082
ER -

References

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