Syntomic cohomology and Beilinson’s Tate conjecture for 𝐾₂
Syntomic cohomology and Beilinson’s Tate conjecture for 𝐾₂
We study Beilinson’s Tate conjecture for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 2"><mml:semantics><mml:msub><mml:mi>K</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:annotation encoding="application/x-tex">K_2</mml:annotation></mml:semantics></mml:math></inline-formula>using the theory of syntomic cohomology. As an application, we construct integral indecomposable elements of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 1"><mml:semantics><mml:msub><mml:mi>K</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:annotation encoding="application/x-tex">K_1</mml:annotation></mml:semantics></mml:math></inline-formula>of elliptic surfaces. Moreover, we give the first example of a surface<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding="application/x-tex">X</mml:annotation></mml:semantics></mml:math></inline-formula>with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" …