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.

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

Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound, i.e. ap(E) = ± 2 √ p . Assuming that all the symmetric power L-functions associated to E have analytic continuation for all s ∈ C, satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner [RT17], and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where ap(E) is fixed because of the Sato-Tate distribution.

We prove central limit theorems (under suitable growth conditions) for sums of quadratic characters, families of Hecke eigenforms of level 1 1 and weight k k , and families of elliptic curves, twisted by an L L -function satisfying certain properties. As a corollary, we obtain a central limit theorem for products χ ( p ) a f ( p ) \chi (p)a_f(p) where χ \chi is a quadratic Dirichlet character and f f is a normalized Hecke eigenform.

We establish an error term in the Sato–Tate theorem of Birch. That is, for$p$prime,$q=p^{r}$and an elliptic curve$E:y^{2}=x^{3}+ax+b$, we show that$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$for any interval$I\subseteq [0,\unicode[STIX]{x1D70B}]$, where the quantity$\unicode[STIX]{x1D703}_{a,b}$is defined by$2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$and$\unicode[STIX]{x1D707}_{ST}(I)$denotes the Sato–Tate measure of the interval$I$.