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The second and third-named authors (arXiv:1705.04115) established a CentralLimit Theorem for the error term in the Sato-Tate law for families of modularforms. This method was adapted to families of elliptic curves in by the firstand second-named authors (arXiv:1705.09229). In this context, a Central LimitTheorem was established only under a strong hypothesis going beyond the RiemannHypothesis. In the present paper, we consider a smoothed version of theSato-Tate conjecture, which allows us to overcome several limitations. Inparticular, for the smoothed version, we are able to establish a Central LimitTheorem for much smaller families of modular forms, and we succeed in proving atheorem of this type for families of elliptic curves under the RiemannHypothesis for $L$-functions associated to Hecke eigenforms for the fullmodular group.

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