A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms

Abstract

Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript k"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">E_{k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the standard fundamental domain for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lie on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A colon equals StartSet e Superscript i theta Baseline colon StartFraction pi Over 2 EndFraction less-than-or-equal-to theta less-than-or-equal-to StartFraction 2 pi Over 3 EndFraction EndSet"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi> θ </mml:mi> </mml:mrow> </mml:msup> <mml:mo>:</mml:mo> <mml:mfrac> <mml:mi> π </mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo> ≤ </mml:mo> <mml:mi> θ </mml:mi> <mml:mo> ≤ </mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A:= \{ e^{i \theta } : \frac {\pi }{2} \leq \theta \leq \frac {2\pi }{3} \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Using this result we prove a speculation of Ono, namely that the zeros of the unique “gap function" in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript k"> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">M_{k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the modular form with the maximal number of consecutive zero coefficients in its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -expansion following the constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , has zeros only on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In addition, we show that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .