Euler numbers in A^1-homotopy theory

(jt. with Kirsten Wickelgren) Given a cohomology theory for (smooth) algebraic varieties, we explain how to use the motivic six functors formalism to associate Euler classes (and Euler numbers) to vector bundles over smooth (and proper) varieties, valued in the cohomology theory. Using more properties of the six functors formalism, in the presence of a non-degenerate section we can compute the Euler number in terms of certain local contributions around the zeros, called indices. We then relate the indices to certain A^1-degrees, and also to the so-called Scheja-Storch form. This generalizes, but is independent of, other results of Kass-Wickelgren.