On the real zeroes of half-integral weight Hecke cusp forms

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>Γ</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mrow><mml:mrow><mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>\</mml:mo> <mml:mi>H</mml:mi></mml:mrow> </mml:mrow> </mml:math> near a cusp at infinity. In analogue of the Ghosh-Sarnak conjecture for classical holomorphic Hecke cusp forms, one expects that almost all of the zeroes sufficiently close to this cusp lie on two vertical geodesics <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>Re</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>Re</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> as the weight tends to infinity. We show that, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mo>≫</mml:mo> <mml:mi>ε</mml:mi></mml:msub> <mml:msup><mml:mi>K</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mo>/</mml:mo> <mml:msup><mml:mrow><mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi></mml:mrow> </mml:msup> </mml:mrow> </mml:math> of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large parameter <i>K</i>, the number of such "real" zeroes grows almost at the expected rate. We also obtain a weaker lower bound for the number of real zeroes that holds for a positive proportion of forms. One of the key ingredients is the estimation of averaged first and second moments of quadratic twists of modular <i>L</i>-functions.