Irreducibility of Hecke polynomials

Murty, M. Ram; Baba, Srinath

doi:10.4310/mrl.2003.v10.n5.a13

Cited by 12 publications

(15 citation statements)

References 8 publications

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“…We begin with a lemma which has been recorded in various forms in [3], [4], and [1]. We give a proof here for completeness.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

On the irreducibility of Hecke polynomials

Ahlgren

2008

Math. Comp.

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Abstract. Let T n,k (X) be the characteristic polynomial of the nth Hecke operator acting on the space of cusp forms of weight k for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials T n,k (X). Using this criterion with some machine computation, we show that if there exists n ≥ 2 such that T n,k (X) is irreducible and has the full symmetric group as Galois group, then the same is true of T p,k (X) for each prime p ≤ 4, 000, 000.

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“…We begin with a lemma which has been recorded in various forms in [3], [4], and [1]. We give a proof here for completeness.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

On the irreducibility of Hecke polynomials

Ahlgren

2008

Math. Comp.

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“…This extends Theorem 1 of [JO98] and Theorem 1.1 of [BM03]. Both theorems restrict to the case N = 1 and assume that there is a unique Galois orbit of newforms, i.e., a unique P, so that no localization is needed.…”

Section: Lemma 1 the Field F F Is Totally Real And Q(amentioning

confidence: 81%

“…The set of p such that a p ∈ Q also has density 1 4 . In the literature there are related but weaker results concerning Corollary 1, which are situated in the context of Maeda's conjecture, i.e., they concern the case of level 1 and assume that the space S k (1) of cusp forms of weight k and level 1 consists of a single Galois orbit of newforms (see, e.g., [JO98] and [BM03]). We now show how Corollary 1 extends the principal results of these two papers.…”

Section: Lemma 1 the Field F F Is Totally Real And Q(amentioning

confidence: 99%

On the generation of the coefficient field of a newform by a single Hecke eigenvalue

Koo¹,

Stein²,

Wiese³

2008

J. Théor. Nombres Bordeaux

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Let f be a non-CM newform of weight k ≥ 2. Let L be a subfield of the coefficient field of f . We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is determined by the inner twists of f . As a particular case, we obtain that in the absence of non-trivial inner twists, the density is 1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.Mathematics Subject Classification (2000): 11F30 (primary); 11F11, 11F25, 11F80, 11R45 (secondary).

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“…Let k be a positive integer, and let l be a prime number. The Hecke polynomial of T Some progress has been made towards this conjecture; methods introduced in [4] prove that certain Hecke polynomials are irreducible and have full Galois group, and results such as those in [6], [2] and [1] show that if a certain T l is irreducible then other T r must be irreducible also.…”

Section: Introductionmentioning

confidence: 99%

“…In [7] and [2], the authors first show that certain of the T 1,1 2,k are irreducible and then using effective versions of the Chebotarev density theorem to show that a positive proportion of the T 1,1 l,k are irreducible, and in [1], plausible hypotheses on the non-vanishing of certain coefficients of modular forms are used to prove irreducibility of the Hecke polynomials (when the weight is 12, for instance, these hypotheses are exactly Lehmer's conjecture on the Ramanujan τ -function).…”

Section: Introductionmentioning

confidence: 99%