Friedberg–Jacquet proved that if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> is a cuspidal automorphic representation of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GL}_{2n}(\textbf{A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>GL</mtext> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> is a functorial transfer from <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GSpin}_{2n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>GSpin</mtext> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> if and only if a global zeta integral <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$Z_H$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Z</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$H = \textrm{GL}_n \times \textrm{GL}_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>=</mo> <msub> <mtext>GL</mtext> <mi>n</mi> </msub> <mo>×</mo> <msub> <mtext>GL</mtext> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is non-vanishing on <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>. We conjecture a p-refined analogue: that any P-parahoric p-refinement <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq10.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\tilde{\pi }^P$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <mi>π</mi> <mo stretchy="false">~</mo> </mover> <mi>P</mi> </msup> </math></EquationSource> </InlineEquation> is a functorial transfer from <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GSpin}_{2n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>GSpin</mtext> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> if and only if a P-twisted version of <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$Z_H$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Z</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation> is non-vanishing on the <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq10.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\tilde{\pi }^P$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <mi>π</mi> <mo stretchy="false">~</mo> </mover> <mi>P</mi> </msup> </math></EquationSource> </InlineEquation>-eigenspace in <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>. This twisted <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$Z_H$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Z</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation> appears in all constructions of p-adic L-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_631_Article_IEq16.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{GL}_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>GL</mtext> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> eigenvariety, and—by proving upper bounds on the dimensions of such families—obtain various results towards the conjecture.

On p-refined Friedberg–Jacquet integrals and the classical symplectic locus in the ${{\,\textrm{GL}\,}}_{2n}$ eigenvariety

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2025年4月25日

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Daniel Barrera Salazar,
Andrew Graham,
Chris Williams

摘要

Friedberg–Jacquet proved that if $\pi $ $π$ is a cuspidal automorphic representation of $\textrm{GL}_{2n}(\textbf{A})$ ${GL}_{2 n} (A)$ , then $\pi $ $π$ is a functorial transfer from $\textrm{GSpin}_{2n+1}$ ${GSpin}_{2 n + 1}$ if and only if a global zeta integral $Z_H$ $Z_{H}$ over $H = \textrm{GL}_n \times \textrm{GL}_n$ $H = {GL}_{n} \times {GL}_{n}$ is non-vanishing on $\pi $ $π$ . We conjecture a p-refined analogue: that any P-parahoric p-refinement $\tilde{\pi }^P$ ${\tilde{π}}^{P}$ is a functorial transfer from $\textrm{GSpin}_{2n+1}$ ${GSpin}_{2 n + 1}$ if and only if a P-twisted version of $Z_H$ $Z_{H}$ is non-vanishing on the $\tilde{\pi }^P$ ${\tilde{π}}^{P}$ -eigenspace in $\pi $ $π$ . This twisted $Z_H$ $Z_{H}$ appears in all constructions of p-adic L-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the $\textrm{GL}_{2n}$ ${GL}_{2 n}$ eigenvariety, and—by proving upper bounds on the dimensions of such families—obtain various results towards the conjecture.

On p-refined Friedberg–Jacquet integrals and the classical symplectic locus in the \({{\,\textrm{GL}\,}}_{2n}\) eigenvariety

摘要