The zeros of even period polynomials for newforms on $\Gamma _0(N)$
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- by SoYoung Choi
- Proc. Amer. Math. Soc. 152 (2024), 909-923
- DOI: https://doi.org/10.1090/proc/16595
- Published electronically: December 18, 2023
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Abstract:
We prove that for even integer $k\geq k_0$, almost all of zeros of the even period polynomial associated to a newform of weight $k$ on $\Gamma _0(N)$ are on the circle $|z|=1/\sqrt {N}$.References
- Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 434929
- SoYoung Choi, The zeros of odd period polynomials for newforms on $\Gamma _0(2)$, Ramanujan J. 62 (2023), no. 3, 761–779. MR 4655148, DOI 10.1007/s11139-023-00705-5
- John Brian Conrey, David W. Farmer, and Özlem Imamoglu, The nontrivial zeros of period polynomials of modular forms Lie on the unit circle, Int. Math. Res. Not. IMRN 20 (2013), 4758–4771. MR 3118875, DOI 10.1093/imrn/rns183
- Ahmad El-Guindy and Wissam Raji, Unimodularity of zeros of period polynomials of Hecke eigenforms, Bull. Lond. Math. Soc. 46 (2014), no. 3, 528–536. MR 3210708, DOI 10.1112/blms/bdu007
- Seokho Jin, Wenjun Ma, Ken Ono, and Kannan Soundararajan, Riemann hypothesis for period polynomials of modular forms, Proc. Natl. Acad. Sci. USA 113 (2016), no. 10, 2603–2608. MR 3482847, DOI 10.1073/pnas.1600569113
- Marvin I. Knopp, Some new results on the Eichler cohomology of automorphic forms, Bull. Amer. Math. Soc. 80 (1974), 607–632. MR 344454, DOI 10.1090/S0002-9904-1974-13520-2
- W. Kohnen and D. Zagier, Modular forms with rational periods, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 197–249. MR 803368
- Vicenţiu Paşol and Alexandru A. Popa, Modular forms and period polynomials, Proc. Lond. Math. Soc. (3) 107 (2013), no. 4, 713–743. MR 3108829, DOI 10.1112/plms/pdt003
- Don Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991), no. 3, 449–465. MR 1106744, DOI 10.1007/BF01245085
Bibliographic Information
- SoYoung Choi
- Affiliation: Department of Mathematics Education and RINS, Gyeongsang National University, 501 Jinjudae-ro, Jinju 52828, Republic of Korea
- MR Author ID: 754586
- Email: mathsoyoung@gnu.ac.kr
- Received by editor(s): February 2, 2023
- Received by editor(s) in revised form: May 18, 2023, and June 20, 2023
- Published electronically: December 18, 2023
- Additional Notes: The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C1A01007112).
- Communicated by: Ling Long
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 909-923
- MSC (2020): Primary 11F03, 11F11
- DOI: https://doi.org/10.1090/proc/16595
- MathSciNet review: 4693655