We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma _0(4)\backslash \mathbb H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="true">\</mo> <mi mathvariant="double-struck">H</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\text {Re}(s)=-1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Re</mtext> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\text {Re}(s)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Re</mtext> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> as the weight tends to infinity. We show that, for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gg _\varepsilon K^2/(\log K)^{3/2+\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>≫</mo> <mi>ε</mi> </msub> <msup> <mi>K</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large parameter K, 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 L-functions.

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

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2026年2月18日

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Jesse Jääsaari

摘要

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold $\Gamma _0(4)\backslash \mathbb H$ $Γ_{0} (4) \ H$ 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 $\text {Re}(s)=-1/2$ $Re (s) = - 1 / 2$ and $\text {Re}(s)=0$ $Re (s) = 0$ as the weight tends to infinity. We show that, for $\gg _\varepsilon K^2/(\log K)^{3/2+\varepsilon }$ $≫_{ε} K^{2} / {(log K)}^{3 / 2 + ε}$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large parameter K, 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 L-functions.