Signs of Fourier coefficients of half-integral weight modular forms

Abstract

Abstract Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c ( n ) denote the n th Fourier coefficient of g , normalized so that c ( n ) is real for all $$n \ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . A theorem of Waldspurger determines the magnitude of c ( n ) at fundamental discriminants n by establishing that the square of c ( n ) is proportional to the central value of a certain L -function. The signs of the sequence c ( n ) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that $$c(n) &lt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and respectively $$c(n) &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> holds for a positive proportion of fundamental discriminants n . Moreover we show that the sequence $$\{c(n)\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>c</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c ( n ) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4 N with N odd, square-free.

Locations

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