Journal of the Ramanujan Mathematical Society

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A Trace Formula for Finite Upper Half Planes


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1 Department of Mathematics, California State University, 5151 State University Drive, Los Angeles-90032, California, United States
     

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In this paper, we prove a trace formula for finite upper half planes Hq. A brief outline is as follows: Fix a subgroup Γ⊂GL(2,𝔽q). The adjacency operators Aa act on functions in L2(Γ\Hq); thus, we may consider AΓa= Aa|L2(Γ\Hq).We prove a trace formula which is an equality between a weighted sum of the traces of the operators AΓa and a sum over the conjugacy classes of Γ. The trace formula allows us to compute the trace of AΓa. We compute the trace formula for the subgroups Γ=N and Γ=K of GL(2, 𝔽q).
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  • A Trace Formula for Finite Upper Half Planes

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Authors

Anthony Shaheen
Department of Mathematics, California State University, 5151 State University Drive, Los Angeles-90032, California, United States

Abstract


In this paper, we prove a trace formula for finite upper half planes Hq. A brief outline is as follows: Fix a subgroup Γ⊂GL(2,𝔽q). The adjacency operators Aa act on functions in L2(Γ\Hq); thus, we may consider AΓa= Aa|L2(Γ\Hq).We prove a trace formula which is an equality between a weighted sum of the traces of the operators AΓa and a sum over the conjugacy classes of Γ. The trace formula allows us to compute the trace of AΓa. We compute the trace formula for the subgroups Γ=N and Γ=K of GL(2, 𝔽q).