On Fourier coefficients of Maass waveforms for 𝑃𝑆𝐿(2,𝑍)

Abstract

In this paper, we use machine experiments to test the validity of the Sato-Tate conjecture for Maass waveforms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper P normal upper S normal upper L left-parenthesis 2 comma m a t h b b upper Z right-parenthesis minus upper H"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">P</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>m</mml:mi> <mml:mi>a</mml:mi> <mml:mi>t</mml:mi> <mml:mi>h</mml:mi> <mml:mi>b</mml:mi> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {PSL}(2,\ mathbb{Z})\backslash H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also elaborate on Stark’s iterative method for calculating the Fourier coefficients of such forms.

Locations

• Mathematics of Computation - View - PDF