<p>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" alttext="p Subscript g Baseline not-equals 0"><mml:semantics><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo>≠<!-- ≠ --></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">p_g\ne 0</mml:annotation></mml:semantics></mml:math></inline-formula>over a<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-adic field such that the torsion of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C normal upper H Subscript 0 Baseline left-parenthesis upper X right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">H</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {CH}_0(X)</mml:annotation></mml:semantics></mml:math></inline-formula>is finite.</p>