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<p>In this paper, we establish the local converse theorem and the stability of local gamma factors for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\widetilde{\textrm{Sp}}_{2n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mtext>Sp</mtext> <mo stretchy="true">~</mo> </mover> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> via the precise local theta correspondence between <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\widetilde{\textrm{Sp}}_{2n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mtext>Sp</mtext> <mo stretchy="true">~</mo> </mover> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\textrm{SO}_{2n+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>SO</mtext> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> over local fields of characteristic not equal to 2. We also prove the rigidity theorem for irreducible generic cuspidal automorphic representations of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\widetilde{\textrm{Sp}}_{2n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mtext>Sp</mtext> <mo stretchy="true">~</mo> </mover> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> over number fields.</p>

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The local converse theorem for \(\widetilde{{\textrm{Sp}}}_{2n}\)

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  • 发表日期：

    2025年12月1日

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  • Jaeho Haan

摘要

In this paper, we establish the local converse theorem and the stability of local gamma factors for \(\widetilde{\textrm{Sp}}_{2n}\) Sp ~ 2 n via the precise local theta correspondence between \(\widetilde{\textrm{Sp}}_{2n}\) Sp ~ 2 n and \(\textrm{SO}_{2n+1}\) SO 2 n + 1 over local fields of characteristic not equal to 2. We also prove the rigidity theorem for irreducible generic cuspidal automorphic representations of \(\widetilde{\textrm{Sp}}_{2n}\) Sp ~ 2 n over number fields.