Large sieve inequalities for periods of Maass forms

Abstract For $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> a Fuchsian Group of the first kind, we obtain large sieve inequalities with weights the hyperbolic periods of Maass forms of even weight. This is inspired by work of Chamizo, who proved a large sieve inequality with weights values of Maass forms of weight 0. The motivation is applications in counting problems in $$\Gamma _1 \backslash \Gamma /\Gamma _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>\</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> , where $$\Gamma _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , $$\Gamma _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> are hyperbolic subgroups of $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> .