Slopes of modular forms and congruences

Douglas L. Ulmer

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 1, page 1-32
  • ISSN: 0373-0956

Abstract

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Our aim in this paper is to prove congruences between on the one hand certain eigenforms of level p N and weight greater than 2 and on the other hand twists of eigenforms of level p N and weight 2. One knows a priori that such congruences exist; the novelty here is that we determine the character of the form of weight 2 and the twist in terms of the slope of the higher weight form, i.e., in terms of the valuation of its eigenvalue for U p . Curiously, we also find a relation between the leading terms of the p -adic expansions of the eigenvalues for U p of the two forms. This allows us to determine the restriction to the decomposition group at p of the Galois representation modulo p attached to the higher weight form.

How to cite

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Ulmer, Douglas L.. "Slopes of modular forms and congruences." Annales de l'institut Fourier 46.1 (1996): 1-32. <http://eudml.org/doc/75171>.

@article{Ulmer1996,
abstract = {Our aim in this paper is to prove congruences between on the one hand certain eigenforms of level $pN$ and weight greater than 2 and on the other hand twists of eigenforms of level $pN$ and weight 2. One knows a priori that such congruences exist; the novelty here is that we determine the character of the form of weight 2 and the twist in terms of the slope of the higher weight form, i.e., in terms of the valuation of its eigenvalue for $U_p$. Curiously, we also find a relation between the leading terms of the $p$-adic expansions of the eigenvalues for $U_p$ of the two forms. This allows us to determine the restriction to the decomposition group at $p$ of the Galois representation modulo $p$ attached to the higher weight form.},
author = {Ulmer, Douglas L.},
journal = {Annales de l'institut Fourier},
keywords = {congruences between modular forms; slopes of modular forms; Galois representations},
language = {eng},
number = {1},
pages = {1-32},
publisher = {Association des Annales de l'Institut Fourier},
title = {Slopes of modular forms and congruences},
url = {http://eudml.org/doc/75171},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Ulmer, Douglas L.
TI - Slopes of modular forms and congruences
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 1
SP - 1
EP - 32
AB - Our aim in this paper is to prove congruences between on the one hand certain eigenforms of level $pN$ and weight greater than 2 and on the other hand twists of eigenforms of level $pN$ and weight 2. One knows a priori that such congruences exist; the novelty here is that we determine the character of the form of weight 2 and the twist in terms of the slope of the higher weight form, i.e., in terms of the valuation of its eigenvalue for $U_p$. Curiously, we also find a relation between the leading terms of the $p$-adic expansions of the eigenvalues for $U_p$ of the two forms. This allows us to determine the restriction to the decomposition group at $p$ of the Galois representation modulo $p$ attached to the higher weight form.
LA - eng
KW - congruences between modular forms; slopes of modular forms; Galois representations
UR - http://eudml.org/doc/75171
ER -

References

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  12. [Sc] A. J. SCHOLL, Motives for modular forms, Invent. Math., 100 (1990), 419-430. Zbl0760.14002MR91e:11054
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