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<p>In this article we will give a formula for the central value of the completed <i>L</i>-function <i>L</i>(<i>s</i>, Sym<sup>2</sup><i>g</i> × <i>f</i>), where <i>f</i> and <i>g</i> are Hilbert newforms, by explicitly computing the local integrals appearing in the refined Gan–Gross–Prasad formula for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\rm SL}_{2} \times {\widetilde {\rm SL}_{2}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>×</mo> <mrow> <msub> <mrow> <mover> <mrow> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo>∼</mo> </mover> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>. We also work out the rationality of this value in some special cases and give a conjecture for the general case.</p>

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Central values of degree six L-functions: the case of Hilbert modular forms

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  • 发表日期：

    2025年7月30日

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  • Utkarsh Agrawal

摘要

In this article we will give a formula for the central value of the completed L-function L(s, Sym2g × f), where f and g are Hilbert newforms, by explicitly computing the local integrals appearing in the refined Gan–Gross–Prasad formula for \({\rm SL}_{2} \times {\widetilde {\rm SL}_{2}}\) S L 2 × S L ∼ 2 . We also work out the rationality of this value in some special cases and give a conjecture for the general case.