<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$E_k(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the normalized Eisenstein series of weight <i>k</i> for the full modular group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\text {SL}(2, \mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that all the zeros of the cusp form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$E_k(z)E_\ell (z)-E_{k+\ell }(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>E</mi> <mi>ℓ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>E</mi> <mrow> <mi>k</mi> <mo>+</mo> <mi>ℓ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the standard fundamental domain lie on the boundary.</p>

Zeros of cuspidal projections of products of Eisenstein series

期刊论文
发表日期：

2025年11月18日

查看全文

Hui Xue,
Daozhou Zhu

摘要

Let $E_k(z)$ $E_{k} (z)$ be the normalized Eisenstein series of weight k for the full modular group $\text {SL}(2, \mathbb {Z})$ $SL (2, Z)$ . We show that all the zeros of the cusp form $E_k(z)E_\ell (z)-E_{k+\ell }(z)$ $E_{k} (z) E_{ℓ} (z) - E_{k + ℓ} (z)$ in the standard fundamental domain lie on the boundary.