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• W2015017002 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<!-- 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<!-- θ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo>:</mml:mo> <mml:mfrac> <mml:mi>π<!-- π --></mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>θ<!-- θ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π<!-- 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 {2pi }{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>." @default.
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• W2015017002 date "2004-03-04" @default.
• W2015017002 modified "2023-10-14" @default.
• W2015017002 title "A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms" @default.
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• W2015017002 doi "https://doi.org/10.1090/s0002-9939-04-07478-7" @default.
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