Sign changes of coefficients of half-integral weight Hecke eigenforms
HTML articles powered by AMS MathViewer
- by Chenran Xu
- Proc. Amer. Math. Soc. 151 (2023), 4633-4642
- DOI: https://doi.org/10.1090/proc/16527
- Published electronically: August 4, 2023
- HTML | PDF | Request permission
Abstract:
Let $\mathfrak {f}$ be a cusp form of half-weight $k+1/2$ and at most quadratic nebentype character whose Fourier coefficients are denoted as $\mathfrak {a}_\mathfrak {f}(n)$. We study sign changes of the family $\{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}}$ where $t$ is a square-free number and $p$ runs through the prime numbers. By taking use of the relationship between the cusp form $\mathfrak {f}$ of half-integral weight and the Shimura lift $f_t$, we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence $\{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}}$.References
- Jan Hendrik Bruinier and Winfried Kohnen, Sign changes of coefficients of half integral weight modular forms, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 57–65. MR 2512356, DOI 10.1017/CBO9780511543371.005
- Winfried Kohnen, A short note on Fourier coefficients of half-integral weight modular forms, Int. J. Number Theory 6 (2010), no. 6, 1255–1259. MR 2726580, DOI 10.1142/S1793042110003484
- Shinji Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1975), 147–161. MR 364106, DOI 10.1017/S0027763000016445
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- W. Kohnen, Y.-K. Lau, and J. Wu, Fourier coefficients of cusp forms of half-integral weight, Math. Z. 273 (2013), no. 1-2, 29–41. MR 3010150, DOI 10.1007/s00209-012-0994-z
- E. Kowalski, Y.-K. Lau, K. Soundararajan, and J. Wu, On modular signs, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 3, 389–411. MR 2726725, DOI 10.1017/S030500411000040X
- Kaisa Matomäki, On signs of Fourier coefficients of cusp forms, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 2, 207–222. MR 2887873, DOI 10.1017/S030500411100034X
- Winfried Kohnen, Modular forms of half-integral weight on $\Gamma _{0}(4)$, Math. Ann. 248 (1980), no. 3, 249–266. MR 575942, DOI 10.1007/BF01420529
- Winfried Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. MR 660784, DOI 10.1515/crll.1982.333.32
- Dinakar Ramakrishnan, Modularity of the Rankin-Selberg $L$-series, and multiplicity one for $\textrm {SL}(2)$, Ann. of Math. (2) 152 (2000), no. 1, 45–111. MR 1792292, DOI 10.2307/2661379
- James Newton and Jack A. Thorne, Symmetric power functoriality for holomorphic modular forms, Publ. Math. Inst. Hautes Études Sci. 134 (2021), 1–116. MR 4349240, DOI 10.1007/s10240-021-00127-3
- C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752, DOI 10.24033/asens.1595
- Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
- J. Cogdell and P. Michel, On the complex moments of symmetric power $L$-functions at $s=1$, Int. Math. Res. Not. 31 (2004), 1561–1617. MR 2035301, DOI 10.1155/S1073792804132455
- Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0
- Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 29–98. MR 2827723, DOI 10.2977/PRIMS/31
Bibliographic Information
- Chenran Xu
- Affiliation: School of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
- Received by editor(s): June 10, 2022
- Received by editor(s) in revised form: October 26, 2022, January 20, 2023, and March 23, 2023
- Published electronically: August 4, 2023
- Communicated by: Amanda Folsom
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4633-4642
- MSC (2020): Primary 11F30, 11F41, 11F03, 11E45, 11-04
- DOI: https://doi.org/10.1090/proc/16527
- MathSciNet review: 4634869