On congruences mod ${\mathfrak p}^m$ between eigenforms and their attached Galois representations

Imin Chen; Ian Kiming; Jonas B. Rasmussen

On congruences mod ${\mathfrak p}^m$ between eigenforms and their attached Galois representations

Imin Chen, Ian Kiming, Jonas B. Rasmussen

Department of Mathematical Sciences

6 Citations (Scopus)

Abstract

Given a prime p and cusp forms f₁ and f₂ on some Γ₁ (N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f₁ and f₂ have the same eigenvalues mod p^m (where p is a fixed prime of K over p) for Hecke operators T_ℓ at all primes ℓ {does not divide} N p. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod p^m in the above sense. The prime motivation for this study is the connection to modular mod p^m Galois representations, and we also explain this connection.

Original language	English
Journal	Journal of Number Theory
Volume	130
Pages (from-to)	608-619
Number of pages	12
ISSN	0022-314X
Publication status	Published - Mar 2010

Cite this

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title = "On congruences mod ${\mathfrak p}^m$ between eigenforms and their attached Galois representations",

abstract = "Given a prime p and cusp forms f1 and f2 on some Γ1 (N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f1 and f2 have the same eigenvalues mod pm (where p is a fixed prime of K over p) for Hecke operators Tℓ at all primes ℓ {does not divide} N p. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod pm in the above sense. The prime motivation for this study is the connection to modular mod pm Galois representations, and we also explain this connection.",

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T1 - On congruences mod ${\mathfrak p}^m$ between eigenforms and their attached Galois representations

AU - Chen, Imin

AU - Kiming, Ian

AU - Rasmussen, Jonas B.

PY - 2010/3

Y1 - 2010/3

N2 - Given a prime p and cusp forms f1 and f2 on some Γ1 (N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f1 and f2 have the same eigenvalues mod pm (where p is a fixed prime of K over p) for Hecke operators Tℓ at all primes ℓ {does not divide} N p. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod pm in the above sense. The prime motivation for this study is the connection to modular mod pm Galois representations, and we also explain this connection.

AB - Given a prime p and cusp forms f1 and f2 on some Γ1 (N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f1 and f2 have the same eigenvalues mod pm (where p is a fixed prime of K over p) for Hecke operators Tℓ at all primes ℓ {does not divide} N p. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod pm in the above sense. The prime motivation for this study is the connection to modular mod pm Galois representations, and we also explain this connection.

M3 - Journal article

SN - 0022-314X

VL - 130

SP - 608

EP - 619

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

On congruences mod ${\mathfrak p}^m$ between eigenforms and their attached Galois representations

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