On the number of dominating Fourier coefficients of two newforms

Chiriac, Liubomir

doi:10.1090/proc/14145

On the number of dominating Fourier coefficients of two newforms
HTML articles powered by AMS MathViewer

by Liubomir Chiriac PDF

Proc. Amer. Math. Soc. 146 (2018), 4221-4224 Request permission

Abstract:

Let $f\!=\!\sum _{n\geq 1} \lambda _f(n)n^{(k_1-1)/2}q^n$ and $g\!=\!\sum _{n\geq 1} \lambda _g(n)n^{(k_2-1)/2}q^n$ be two newforms with real Fourier coeffcients. If $f$ and $g$ do not have complex multiplication and are not related by a character twist, we prove that \begin{equation*} \#\{n\leq x~|~\lambda _f(n)>\lambda _g(n)\}\gg x. \end{equation*}

References

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
Liubomir Chiriac, Comparing Hecke eigenvalues of newforms, Arch. Math. (Basel) 109 (2017), no. 3, 223–229. MR 3687866, DOI 10.1007/s00013-017-1072-x
Michael Harris, Galois representations, automorphic forms, and the Sato-Tate conjecture, Indian J. Pure Appl. Math. 45 (2014), no. 5, 707–746. MR 3286083, DOI 10.1007/s13226-014-0085-4
Michael Harris, Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston, Boston, MA, 2009, pp. 1–21. MR 2641185, DOI 10.1007/978-0-8176-4747-6_{1}
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
Emmanuel Kowalski, Olivier Robert, and Jie Wu, Small gaps in coefficients of $L$-functions and $\mathfrak {B}$-free numbers in short intervals, Rev. Mat. Iberoam. 23 (2007), no. 1, 281–326. MR 2351136, DOI 10.4171/RMI/496
Y.-K. Lau and J. Wu, The number of Hecke eigenvalues of same signs, Math. Z. 263 (2009), no. 4, 959–970. MR 2551607, DOI 10.1007/s00209-008-0448-9
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
M. Ram Murty and Sudhir Pujahari, Distinguishing Hecke eigenforms, Proc. Amer. Math. Soc. 145 (2017), no. 5, 1899–1904. MR 3611306, DOI 10.1090/proc/13446