# Slopes of modular forms and congruences

Annales de l'institut Fourier (1996)

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

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topUlmer, 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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