Correction to: Equidistribution of Eisenstein Series in the Level Aspect
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Communications in
Mathematical Physics
Correction
Correction to: Equidistribution of Eisenstein Series in the Level Aspect Ikuya Kaneko1 , Shin-ya Koyama2 1 Tsukuba Kaisei High School, 3315-10 Kashiwada, Ushiku, Ibaraki 300-1211, Japan.
E-mail: [email protected]
2 Department of Biomedical Engineering, Toyo University, 2100 Kujirai, Kawagoe,
Saitama 350-8585, Japan. E-mail: [email protected] Received: 12 October 2019 / Accepted: 8 May 2020 Published online: 10 October 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Correction to: Commun. Math. Phys. https://doi.org/10.1007/s00220-009-0764-x The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in “Equidistribution of Eisenstein series in the level aspect”, Commun. Math. Phys. 289(3), 1131–1150 (2009). We point out errors and correct the proofs with partially weakened claims. This article of corrigendum and addendum comprises three sections. Section 1 is devoted to making concise corrections to the claim and the proof of Theorems 1.2 and 1.3, and Sect. 2 serves as a modification of the computation of the contribution from Maaß cusp forms. Other relatively marginal mathematical issues and notational/typographical errors are listed afterwards in Sect. 3. 1. Correction to Theorem 1.3 In this section we correct the main theorem (Theorem 1.3). In the original paper [7], Theorem 1.2 was derived as a lucid corollary of Theorem 1.3, while we conclude that the content of Theorem 1.2 ought to hold with a little modification. In our fresh proof, we could no longer rely upon the Luo–Sarnak approach as the level is varying. It behoves us to mention that the quantum unique ergodicity (QUE) conjecture for unitary Eisenstein series in the level aspect actually cannot hold for all systems of test functions whose levels are sufficiently large. This is in contradiction to the results shown in [7], where the test function was supposed to be of level q. We define Eisenstein series indexed by cusps a singular for a nebentypus χ : (q) E a (z, s, χ ) = E a (z, s, χ ) := χ(γ )(σa−1 γ z)s , γ ∈a \0 (q)
The original article can be found online at https://doi.org/10.1007/s00220-009-0764-x.
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which converges absolutely for (s) > 1 and z ∈ H, where a := {γ ∈ 0 (q) : γ a = a} stands for the stabiliser of the cusp a, and the scaling matrix σa ∈ SL2 (R) is such that σa ∞ = a, σa−1 a σa = ∞ . In the following lines, we often drop the superscript (q), and also suppress χ if it is principal. So long as there is no risk of confusion, we simply write the Eisenstein series as E. We here introduce the standard shorthand. Given an integer q ≥ 1 and a central character χ (mod q), denote by Y0 (q) = 0 (q)\H the modular curve and by L 2 (Y0 (q), χ ) the space of measurable functions f : H → C satisfying the periodicity condition az + b ab = χ (d) f (z) for all ∈ 0 (q) f cd cz + d and f, f q < ∞, where ·, · q signifies the Petersson inner product f (z)g(z)dμ(z) f, g q := Y0 (q)
with
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