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Equidistribution of Eisenstein Series in the Level Aspect

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A Correction to this article was published on 10 October 2020

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Abstract

We prove an equidistribution property of the Eisenstein series for congruence subgroups as the level goes to infinity. This is an analogy of the phenomenon called quantum ergodicity.

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Change history

  • 10 October 2020

    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) 1150 (2009). We point out errors and correct the proofs with partially weakened claims.

References

  1. Duke W., Friedlander J.B., Iwaniec H.: The subconvexity problem for Artin L-functions. Invent. Math. 149, 489–577 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Hoffstein J., Lockhart P.: Coefficients of Maass forms and the Siegel zero. Ann. Math. 140, 161–181 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Iwaniec, H.: Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, Vol. 53, Providence, RI: Amer. Math. Soc.

  4. Iwaniec, H., Kowalski, E.: Analytic Number Theory. Colloquium Publications 53, Providence, RI: Amer. Math. Soc., 2004

  5. Koyama S.: Quantum ergodicity of Eisenstein series for arithmetic 3-manifolds. Commun. Math. Phys. 215, 477–486 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Kowalski E., Michel P., Vanderkam J.: Rankin-Selberg L-functions in the level aspect. Duke Math. J. 114, 123–191 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lindenstrass E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. 163, 165–219 (2006)

    Article  MathSciNet  Google Scholar 

  8. Luo W., Sarnak P.: Quantum ergodicity of eigenfunctions on \({PSL_2(\mathbb{Z})\backslash H^2}\) . Publ. I.H.E.S. 81, 207–237 (1995)

    MATH  MathSciNet  Google Scholar 

  9. Ramanujan S.: Some formulae in the arithmetic theory of numbers. Messenger of Math. 45, 81–84 (1916)

    Google Scholar 

  10. Rudnick Z., Sarnak P.: The behavior of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys 161, 195–213 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Meurman, T.: On the order of the Maass L-function on the critical line. In: Number theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai, 51, Amsterdams: North-Holland, 1990, pp. 325–354

  12. Petridis Y., Sarnak P.: Quantum ergodicity for SL(2, o)\H 3 and estimates for L functions. J. Evol. Equ. 1, 277–290 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sarnak P.: Estimates for Rankin-Selberg L-functions and quantum unique ergodicity. J. Funct. Anal. 184, 419–445 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Institute of Mathematical Sciences, Ewha Womans University, Daehyun-dong 11-1, Sedaemoon-ku, 120-750, Seoul, South Korea

    Shin-ya Koyama

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  1. Shin-ya Koyama
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Correspondence to Shin-ya Koyama.

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Communicated by S. Zelditch

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Koyama, Sy. Equidistribution of Eisenstein Series in the Level Aspect. Commun. Math. Phys. 289, 1131–1150 (2009). https://doi.org/10.1007/s00220-009-0764-x

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  • DOI: https://doi.org/10.1007/s00220-009-0764-x

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