On the signs of Fourier coefficients of Hilbert cusp forms
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On the signs of Fourier coefficients of Hilbert cusp forms Ritwik Pal1 Received: 6 December 2018 / Accepted: 19 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove that given any > 0 and a primitive adelic Hilbert cusp form f of weight k = (k1 , k2 , . . . , kn ) ∈ (2Z)n and full level, there exists an integral ideal m with 9/20+ N (m) Q f such that the m-th Fourier coefficient of C f (m) of f is negative. Here n is the degree of the associated number field, N (m) is the norm of integral ideal m and Q f is the analytic conductor of f . In the case of arbitrary weights, we show 1/2+ that there is an integral ideal m with N (m) Q f such that C f (m) < 0. We n also prove that when k = (k1 , k2 , . . . , kn ) ∈ (2Z) , asymptotically half of the Fourier coefficients are positive while half are negative. Keywords Hilbert modular forms · First sign change · Fourier coefficients · Distribution of eigenvalues Mathematics Subject Classification 11F30 · 11F41
1 Introduction This paper is concerned with a quantitative result on the study of signs of Fourier coefficients of Hilbert cusp forms. This theme of research has seen a lot of activity in the recent past; here we just recall the landmark result of Matomäki for elliptic Hecke cusp forms, which also sets the ground of our results to follow. Let Q f k 2 N be the analytic conductor of an elliptic newform f of level N and weight k. Then it was proved in [8] that the first negative eigenvalue of f occurs at some n 0 ≥ 1 with 3/8 n 0 Q f . The method in [8] is based on further refinement of that in [5] wherein a variety of results on statistical distribution of signs of Fourier coefficients of newforms were studied. In the case of Hilbert newforms of arbitrary weight and level, the only known result seems to be the work of Meher and Tanabe (see [10, Theorems 1.1, 1.2]). To describe
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Ritwik Pal [email protected]; [email protected] Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
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their result let us introduce the following notation. Let F be a totally real number field of degree n associated with the adelic Hilbert newform f . Let {C(m)}m denote the Fourier coefficients of f , indexed by the integral ideals m, and Q f denote the analytic conductor of f (see next section for the definition). Then in [10] it is shown that the sequence {C(m)}m changes sign infinitely often and more quantitatively the main result of [10] states that there exists an integral ideal m with N (m) n, Q 1+ f such that C(m) < 0, where N (m) is the norm of integral ideal m. One of the aims of this paper is to improve upon this result. Our main result is the following. Theorem 1.1 Let f be a Hilbert newform of weight k = (k1 , k2 , . . . , kn ) and full level. Let C(m) denote the Fourier coefficient of f at the ideal m. Then for any arbitrary > 0, (i) when k1 , k2 , . . . , kn are all even, we have C(m) < 0 for some ideal m with 9
N (m) n, Q 20 f
+
;
(ii) otherwise we have C(m)
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