Distribution of Hecke eigenvalues

Abstract

We give two results concerning the distribution of Hecke eigenvalues of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis 2 comma double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SL(2, \mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The first result asserts that on certain average the Sato-Tate conjecture holds. The second result deals with the Gaussian central limit theorem for Hecke eigenvalues.