Independence of $\ell $-adic Galois representations over function fields. (July 2013)

• Record Type: Journal Article
• Authors: Gajda, Wojciech; Petersen, Sebastian
• Is Part Of: Compositio mathematica. Volume 149:Number 7(2013)
• Page Start: 1091
• Page End: 1107
• Publication Date: 2013-07
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X12000711
• Languages: English
• ISSNs: 0010-437X

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91f5r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$K$]]></tex-math></alternatives></inline-formula> be a finitely generated extension of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91fsp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbb {Q}$]]></tex-math></alternatives></inline-formula>. We consider the family of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91db0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\ell $]]></tex-math></alternatives></inline-formula>-adic representations (<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91fh8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\ell $]]></tex-math></alternatives></inline-formula> varies through the set of all prime numbers) of the absolute Galois group of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91gs6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$K$]]></tex-math></alternatives></inline-formula>, attached to <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91dmd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\ell $]]></tex-math></alternatives></inline-formula>-adic cohomology of a separated scheme of finite type over <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91cnf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$K$]]></tex-math></alternatives></inline-formula>. We prove that the fields cut out from the algebraic closure of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg91hzg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$K$]]></tex-math></alternatives></inline-formula>by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.</p> </abstract>