On the gaps in the Fourier expansion of cusp forms

Alkan, Emre; Zaharescu, Alexandru

doi:10.1007/s11139-007-9091-z

Cited by 7 publications

(3 citation statements)

References 13 publications

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“…The approaches are either using Rankin-Selberg estimates, or Chebotarev density theorem, or distribution of B-free numbers, etc., (cf. [BO01], [Alk03], [Alk05], [AZ05a], [AZ05b], [Mat12], [KRW07], [AZ08]).…”

Section: Introductionmentioning

confidence: 99%

On the gaps between nonzero Fourier coefficients of eigenforms with CM

Kaushik

Kumar

2017

Int. J. Number Theory

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Suppose E is an elliptic curve over Q of conductor N with complex multiplication (CM) by Q(i), and f E is the corresponding cuspidal Hecke eigenform in S new 2 (Γ 0 (N )). Then n-th Fourier coefficient of f E is non-zero in the short interval (X, X + cX 1 4 ) for all X ≫ 0 and for some c > 0. As a consequence, we produce infinitely many cuspidal CM eigenforms f level N > 1 and weight k > 2 for which i f (n) ≪ n 1 4 holds, for all n ≫ 0.

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Section: Introductionmentioning

confidence: 99%

On the gaps between nonzero Fourier coefficients of eigenforms with CM

Kaushik

Kumar

2017

Int. J. Number Theory

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“…The approaches are either using Rankin-Selberg estimates, or Chebotarev density theorem, or distribution of B-free numbers, or the bounds on certain exponential sums, etc., (cf. [BO01], [Alk03], [Alk05], [AZ05a], [AZ08], [DG14]). For example, the approach to the non-vanishing problem through the distribution of B-free numbers has been considered by Alkan and Zaharescu ([AZ05b]), Matomäki ([Mat12]), and Kowalski, Robert, and Wu ( [KRW07]), and many more.…”

Section: Introductionmentioning

confidence: 99%

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

Kumar

2016

Ramanujan J

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Abstract. We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve E/Q, which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval (X, X + cX 1 4 ) for all X ≫ 0 and for some c > 0. We use this fact to produce non-CM cuspidal eigenforms f of level N > 1 and weight k > 2 such that i f (n) ≪ n 1 4 for all n ≫ 0.

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“…The question remains, nevertheless, of reducing the size of the exponent δ, and several researchers contributed towards answering this question and its several modifications. See, for example, [BO01], [Al03], [Al05], [AZa05], [AZ08], and [KRW07], the last one, due to Kowalski, Robert, and Wu, containing the strongest general result. They use new bounds on certain exponential sums and distribution of B-free numbers to show that i f (n) f n 7/17+ε for any ε > 0 for any cusp form f which is not a linear combination of forms with complex multiplication.…”

Section: Introductionmentioning

confidence: 99%