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

Abstract

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. Our estimates are stronger than those obtained in [4] and [5] for the first and second moment, but this comes at the cost of larger ranges of averaging. 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. Birch's identities connect moments of coefficients of Hasse-Weil L-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak's identity is combinatorial in nature.2010 Mathematics Subject Classification. 11G05 (primary); 11G40 (secondary). 1Here the size of δ depends on ε and is smaller than 1/12 (see equation (20) in [5]).There are a number of related results in the literature (see, in particular, [9] and [21]). In this paper, we treat all moments, not only the first and second moments, and obtain new estimates. Our focus lies on strong savings over the trivial bounds rather than as small as possible families of curves (as weak as possible conditions on A and B), which latter was the goal in the papers [4] and [5] as well as subsequent papers on this subject. Our savings for the first and second moments are indeed stronger than those obtained in [4] and [5]. In particular, for the first moment, we get, for fixed I, a saving of x 1/4 (log x) c unconditionally and x 1/2−ε under MRH (a particular case of the Generalized Riemann Hypothesis, stated below) as compared to the power of logarithm saving in Theorem 1.2(i) and the above-mentioned saving of x δ with δ < 1/12 obtained in [5]. The price of this improvement will be that our families of curves are larger, i.e., our conditions on A and B

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

“…Using the Cauchy-Schwarz inequality, we observe that Lx ε L −1 r−s . Now we continue in a similar way as in [BP19]. However, there are two significant differences between the setting in the present paper and that in [BP19]: Firstly, in [BP19], the coefficients U (m) satisfied the bound U (m) ≪ 1/m, whereas here we have the bound U (m) ≪ 1/L 2 .…”

“…Secondly, in [BP19], we derived our final moment bound under the condition that M ≥π(x) 1/2 , which is not the case here. In the following, we perform the required adjustments to make our method in [BP19] work in the context of this paper.…”

An unconditional explicit bound on the error term in the Sato-Tate conjecture

“…It is expected that a bound of the shape c f,ε x 1 2 +ε is satisfied. Some convincing evidence for this is exhibited in [1] in the elliptic curve case.…”

“…Furthermore, by (2.3), the contributions to ∆ R arg Λ(s, Sym m f ) from the left and right sides of the contour are equal. Accordingly, let C be the part of the contour from 1 2 − iT to…”

“…Proof. Following the example of [6] and [29], we split the contour C into three pieces: C 1 , C 2 , C 3 , corresponding to the line segments connecting 1 2 − iT to σ 1 − iT , σ 1 − iT to σ 1 + iT , and σ 1 + iT to 1 2 + iT , respectively. We first bound the contribution from the line segment C 2 .…”