On an elliptic curve defined by the equation, $$E:y^2=x^3+a x +b$$ The algebraic form $dx/y$ is defined on the elliptic curve and it is a non-vanishing section of the (trivial) canonical bundle. From some physics literature, the form $xdx/y$, which has poles on the elliptic curve, together with $dx/y$ form a basis of the cohomology $H^1(E,\mathbb{C})$, which is not clear to me. Could anyone explain why the form $xdx/y$ defines an element of $H^1(E,\mathbb{C})$?
Edit: For example, in the famous paper by Seiberg and Witten, page 36, the sentence above formula (6.9).