Fluctuations in the Distribution of Hecke Eigenvalues about the Sato–Tate Measure

Abstract

We study fluctuations in the distribution of families of p-th Fourier coefficients a f (p) of normalised holomorphic Hecke eigenforms f of weight k with respect to SL 2 (Z) as k → ∞ and primes p → ∞. These families are known to be equidistributed with respect to the Sato-Tate measure. We consider a fixed interval I ⊂ [−2, 2] and derive the variance of the number of a f (p)'s lying in I as p → ∞ and k → ∞ (at a suitably fast rate). The number of a f (p)'s lying in I is shown to asymptotically follow a Gaussian distribution when appropriately normalised. A similar theorem is obtained for primitive Maass cusp forms.

Moments of the error term in the Sato–Tate law for elliptic curves

“…To achieve these results, we use a method which is different from those in [4] and [5], where the key point was the use of multiplicative characters to detect isomorphism classes of curves modulo primes. Our approach builds instead on the work [22] by the second-named author and K. Sinha about the distribution of the error in the Sato-Tate law for modular forms. Here the starting point is to detect the condition thatã E (p) ∈ I by employing Theorem 1.8 below, which was established in [22] in the context of Fourier coefficients of cusp forms using Beurling-Selberg polynomials and the multiplicative properties of the coefficients in question.…”

“…Our approach builds instead on the work [22] by the second-named author and K. Sinha about the distribution of the error in the Sato-Tate law for modular forms. Here the starting point is to detect the condition thatã E (p) ∈ I by employing Theorem 1.8 below, which was established in [22] in the context of Fourier coefficients of cusp forms using Beurling-Selberg polynomials and the multiplicative properties of the coefficients in question. Then we use identities by Birch which connect moments of the coefficients a E(a,b) (p) with the Kronecker class number and further with traces of Hecke operators (see sections 6 and 7).…”

“…As a second application of Theorem 1.4 the following Central Limit Theorem can be deduced from the last estimate for the moments in (12) under Hypothesis 1 by adapting the method of moments used in [22].…”

“…As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method is different from those used in the above-mentioned papers and builds on recent work by the second-named author and K. Sinha [22] who derived a Central Limit Theorem on the distribution of the errors in the Sato-Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper.…”

Central limit theorems for elliptic curves and modular forms with smooth weight functions

“…We can then evaluate the asymptotic behaviour of expected value of this random variable, the variance and higher moments of suitable normalizations of this random variable. More precisely, [PS17] contains the following theorem.…”