On Zeta Functions of Arithmetically Defined Graphs

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We study the graphX(n) that is defined as the finite part of the quotient Γ(n)\\T, with T the Bruhat–Tits tree over Fq((1/T)) and Γ(n) the principal congruence subgroup of Γ=GL2(Fq[T]) of leveln∈Fq[T]. We give concrete realizations of theL-functions of the finite part of the halfline Γ\\T for finite unitary representations of Γ that factor over Γ(n),nprime. This allows us to give explicit formulae for the zeta function ofX(n) for smalln. As an application, we show that these graphs are very good concentrators. Moreover, we construct a new unbounded family of Ramanujan graphs, considering regularizations ofX(n).