On period relations for automorphic 𝐿-functions I

<p>This paper is the first in a series of two dedicated to the study of period relations of the type <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis one half plus k comma normal upper Pi right-parenthesis element-of left-parenthesis 2 pi i right-parenthesis Superscript d dot k Baseline normal upper Omega Subscript left-parenthesis negative 1 right-parenthesis Sub Superscript k Subscript Baseline reverse-solidus bf upper Q left-parenthesis normal upper Pi right-parenthesis comma one half plus k critical comma"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mspace width="thickmathspace" /> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mspace width="thickmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π<!-- π --></mml:mi> <mml:mi>i</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>\bf Q</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="1em" /> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> <mml:mspace width="thickmathspace" /> <mml:mtext>critical</mml:mtext> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}\textrm {\bf Q}(\Pi ),\quad \frac {1}{2}+k\;\text {critical}, \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> for certain automorphic representations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi"> <mml:semantics> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:annotation encoding="application/x-tex">\Pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a reductive group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G period"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> In this paper we discuss the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals normal upper G normal upper L left-parenthesis n plus 1 right-parenthesis times normal upper G normal upper L left-parenthesis n right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G=\mathrm {GL}(n+1)\times \mathrm {GL}(n).</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals normal upper G normal upper L left-parenthesis 2 n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G=\mathrm {GL}(2n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is discussed in part two. Our method is representation theoretic and relies on the author’s recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi"> <mml:semantics> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:annotation encoding="application/x-tex">\Pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under consideration. The new period relations we prove are in accordance with Deligne’s Conjecture on special values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions, and the author expects this method to apply to other cases as well.</p>