Chow motives associated to certain algebraic Hecke characters

Abstract

Shimura and Taniyama proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a potentially CM abelian variety over a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with CM by a field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> linearly disjoint from F, then there is an algebraic Hecke character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript upper A"> <mml:semantics> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\lambda _A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F upper K"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">FK</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis upper A slash upper F comma s right-parenthesis equals upper L left-parenthesis lamda Subscript upper A Baseline comma s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(A/F,s)=L(\lambda _A,s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We consider a certain converse to their result. Namely, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript e Baseline equals gamma x Superscript f Baseline plus delta"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>e</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>γ<!-- γ --></mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>f</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">y^e=\gamma x^f+\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Fix positive integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n slash 2 greater-than a less-than-or-equal-to n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>a</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n/2 &gt; a \leq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Under mild conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e comma f comma gamma comma delta"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">e, f, \gamma , \delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we construct a Chow motive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F equals double-struck upper Q left-parenthesis gamma comma delta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F=\mathbb {Q}(\gamma ,\delta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis upper M slash upper F comma s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(M/F,s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis lamda Subscript upper A Superscript a Baseline lamda overbar Subscript upper A Superscript n minus a Baseline comma s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>A</mml:mi> <mml:mi>a</mml:mi> </mml:msubsup> <mml:msubsup> <mml:mover> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(\lambda _A^a\overline {\lambda }_A^{n-a},s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have the same Euler factors outside finitely many primes.

Locations

• Transactions of the American Mathematical Society Series B - View - PDF
• arXiv (Cornell University) - View - PDF
• MPG.PuRe (Max Planck Society) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• Oxford University Research Archive (ORA) (University of Oxford) - View - PDF