On the typical size and cancellations among the coefficients of some modular forms

Luca, Florian; Radziwiłł, Maksym; Shparlinski, Igor E.

doi:10.1017/s0305004117000780

Cited by 7 publications

(4 citation statements)

References 26 publications

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“…Now, it is known that there is a constant C f > 0 such that if λ f (p ν ) = 0 then |λ f (p ν )| ≥ p −C f ν . Indeed, as argued in [9], writing a f (n) := λ f (n)n (k−1)/2 for all n we have that a f (n) is an algebraic integer in some number field K f of (finite) degree d f ≥ 1, and so for each n = p ν with a f (n) = 0 its norm N K f (a f (p ν )) ≥ 1. On the other hand, for each automorphism σ of K f it is well known (see [20]) that {σ (a f (n))} n is the sequence of coefficients of an eigencusp form of weight k and level 1, and we obtain…”

Section: And So We Immediately Have Thatmentioning

confidence: 91%

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

Mangerel¹

2023

Math. Proc. Camb. Phil. Soc.

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Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{\lambda_f(n)\}_{n \in \mathcal{N}_K}$ . More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $\lambda_f(n)\lambda_f(n') < 0$ , where n′ is the first element of $\mathcal{N}_K$ greater than n for which $\lambda_f(n') \neq 0$ . For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 \;:\; m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,\varepsilon} X^{1/2-\varepsilon}$ sign changes for $\lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$ .

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Section: And So We Immediately Have Thatmentioning

confidence: 91%

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

Mangerel¹

2023

Math. Proc. Camb. Phil. Soc.

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“…Applying these two results with ∆ = (log X) −c , 0 < c < 1/4, then using partial summation, for any 2 ≤ Y ≤ X we obtain Now, it is known that there is a constant C f > 0 such that if λ f (p ν ) = 0 then |λ f (p ν )| ≥ p −C f ν . Indeed, as argued in [7], writing a f (n) := λ f (n)n (k−1)/2 for all n we have that a f (n) is an algebraic integer in some number field K f of (finite) degree d f ≥ 1, and so for each n = p ν with a f (n) = 0 its norm N K f (a f (p ν )) ≥ 1. On the other hand, for each automorphism σ of K f it is well-known that {a f (n) σ } n is the sequence of coefficients of an eigencusp form of weight k and level 1, and we obtain The second claim is proven more directly by appealing to Lemma 2.4 and partial summation.…”

Section: Proof Of Theorem 13mentioning

confidence: 91%

Sign Changes of Fourier Coefficients of Cusp Forms at Norm Form Arguments

Mangerel¹

2022

Preprint

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Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let {λ f (n)}n be its sequence of normalized Fourier coefficients. We show that if K/Q is any number field, and N K denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers n ∈ N K yield a sign change for the sequenceFor example, for K = Q(i) and N K = {m 2 + n 2 : m, n ∈ Z} the set of sums of two squares, we obtain ≫ f X/ √ log X such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwi l l on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed a = 0 there are ≫ f,ε X 1/2−ε sign changes for λ f along the sequence of integers of the form a + m 2 + n 2 ≤ X.

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“…We next present a sharpening of a result from [14], which shows that the set of primes with λ f (p) = 0 is quite thin.…”

Section: Multiplicative Functions Supported On Thin Setsmentioning

confidence: 98%

“…Remark 1.9. It is known that any non-CM cusp form f satisfies the property b j (p) = 0 ⇒ b f (p ν ) = 0 for all ν for all but finitely many p. As noted on p. 175 of [14], the lower bound b f (p ν ) ≥ p −Cν for some C > 0 holds for all holomorphic Hecke cusp forms, which includes f = ∆. This is why no assumptions are required for the Ramanujan τ function.…”

Section: Introductionmentioning

confidence: 97%

Monotone chains of Fourier coefficients of Hecke cusp forms

Klurman,

Mangerel

2020

Preprint

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We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms f 1 , . . . , f k without complex multiplication, of equal weight, (possibly different) squarefree level and trivial nebentypus. As a first application, we show that for the Ramanujan τ function and any admissible k-tuple of distinct non-negative integers a 1 , . . . , a k the sethas positive natural density. This result improves upon recent work of Bilu, Deshouillers, Gun and Luca [Compos. Math. (2018), no. 11, 2441-2461. Secondly, we make progress towards understanding the signed version by showing that {n ∈ N : τ (n + a 1 ) < τ (n + a 2 ) < τ (n + a 3 )} has positive relative upper density at least 1/6 for any admissible triple of distinct nonnegative integers (a 1 , a 2 , a 3 ). More generally, for such chains of inequalities of length k > 3 we show that under the assumption of Elliott's conjecture on correlations of multiplicative functions, the relative natural density of this set is 1/k!. Previously results of such type were known for k ≤ 2 as consequences of works by Serre and by Matomäki and Radziwi l l. Our results rely crucially on several key ingredients: i) a multivariate Erdős-Kac type theorem for the function n → log |τ (n)|, conditioned on n belonging to the set of non-vanishing of τ , generalizing work of Luca, Radziwi l l and Shparlinski; ii) the recent breakthrough of Newton and Thorne on the functoriality of symmetric power L-functions for GL(n) for all n ≥ 2 and its application to quantitative forms of the Sato-Tate conjecture; and iii) the work of Tao and Teräväinen on the logarithmic Elliott conjecture.

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On the typical size and cancellations among the coefficients of some modular forms

Cited by 7 publications

References 26 publications

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

Sign Changes of Fourier Coefficients of Cusp Forms at Norm Form Arguments

Monotone chains of Fourier coefficients of Hecke cusp forms

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