For even <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\ge 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> and square free <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$D>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$D\equiv 1\pmod 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\chi _D$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>χ</mi> <mi>D</mi> </msub> </math></EquationSource> </InlineEquation> be the primitive quadratic Dirichlet character mod D and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S_k(D,\chi _D)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <msub> <mi>χ</mi> <mi>D</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the space of cusp forms of weight k, level D, and nebentypus <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\chi _D$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>χ</mi> <mi>D</mi> </msub> </math></EquationSource> </InlineEquation>. We show that if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$D>2^{k-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>></mo> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, then the critical values of symmetric square L-functions on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$S_k(D,\chi _D)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <msub> <mi>χ</mi> <mi>D</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are linearly independent.

A note on the critical values of symmetric square L-functions

期刊论文
发表日期：

2025年10月30日

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Tianyu Ni

摘要

For even $k\ge 6$ $k \geq 6$ and square free $D>1$ $D > 1$ with $D\equiv 1\pmod 4$ $D \equiv 1 (mod 4)$ , let $\chi _D$ $χ_{D}$ be the primitive quadratic Dirichlet character mod D and $S_k(D,\chi _D)$ $S_{k} (D, χ_{D})$ be the space of cusp forms of weight k, level D, and nebentypus $\chi _D$ $χ_{D}$ . We show that if $D>2^{k-2}$ $D > 2^{k - 2}$ , then the critical values of symmetric square L-functions on $S_k(D,\chi _D)$ $S_{k} (D, χ_{D})$ are linearly independent.