Quantum Unique Ergodicity for Eisenstein Series in the Level Aspect

We prove a variety of quantum unique ergodicity results for Eisenstein series in the level aspect. A new feature of this variant of QUE is that the main term involves the logarithmic derivative of a Dirichlet $L$-function on the $1$-line. A zero of this $L$-function near the $1$-line can thus have a distorting effect on the main term. We obtain quantitative control on the test function and thereby prove an asymptotic formula in the level aspect version of the problem with test functions of shrinking support. Surprisingly, this asymptotic formula shows some obstruction to equidistribution that may retrospectively be interpreted as being caused by the growth of Eisenstein series in the cusps. We also make some coarse descriptions on the unevenness of the mass distribution of level $N$ Eisenstein series on the fibers of the canonical projection map from $Y_0(N)$ to $Y_0(1)$.

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