Kowalski–Michel–VanderKam mass equidistribution conjecture in the level aspect

Let ff be a classical holomorphic newform of fixed weight kk and level qq, with mass dνf=f(z)2yk2dxdyd\nu_f=|f(z)|^2y^{k-2}\,dx\,dy on Y0(q)=Γ0(q)\HY_0(q)=\Gamma_0(q)\backslash\mathbb H. Let πq:Y0(q)Y0(1)\pi_q:Y_0(q)\rightarrow Y_0(1) be the canonical projection, and let μ\mu denote the Poincaré measure on Y0(1)Y_0(1). Kowalski–Michel–VanderKam's mass equidistribution conjecture. If ff traverses a sequence of newforms of fixed weight and increasing level qq\rightarrow\infty, then the pushforward μf:=πq(νf)\mu_f:=\pi_{q*}(\nu_f) equidistributes with respect to μ\mu. This is the level-aspect analogue of the Rudnick–Sarnak quantum unique ergodicity conjecture, reversing the roles of weight and level; the supplied text does not state whether it has been resolved.

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Primary source

Paul D. Nelson, “Equidistribution of cusp forms in the level aspect”, arXiv:1011.1292 (2012).

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