On interlacing of zeros of certain family of modular forms

2017/02/23 by Ekata Saha, N. Saradha, Saha, Ekata +1
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1702.07296

Abstract

Let k=12 m(k)+s ≥ 12 for s∈ \0,4,6,8,10,14\, be an even integer and f be a normalised modular form of weight k with real Fourier coefficients, written as f=E_k+∑_j=1^m(k)a_jE_k-12jΔ^j. Under suitable conditions on a_j (rectifying an earlier result of Getz), we show that all the zeros of f, in the standard fundamental domain for the action of \bf SL(2,\mathbb Z) on the upper half plane, lies on the arc A:= \ e^{i θ} : \fracπ2 ≤ θ≤ (2π)/(3) \. Further, extending a result of Nozaki, we show that for certain family \f_k\_k of normalised modular forms, the zeros of f_k and f_k+12 interlace on A^∘:= \ e^{i θ} : \fracπ2 < θ< (2π)/(3) \.

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