Beschreibung:
<jats:title>Abstract</jats:title><jats:p>We study a symplectic variant of algebraic <jats:italic>K</jats:italic>-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathbf {Q}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Q</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula>. We compute this action explicitly. The representations we see are extensions of Tate twists <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathbf {Z}}_p(2k-1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mi>Z</mml:mi>
<mml:mi>p</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>k</mml:mi>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties.
</jats:p>