  We give a conditional bound for the average analytic rank of elliptic curves
over an arbitrary number field. In particular, under the assumptions that all
elliptic curves over a number field $K$ are modular and have $L$-functions
which satisfy the Generalized Riemann Hypothesis, we show that the average
analytic rank of isomorphism classes of elliptic curves over $K$ is bounded
above by $3\text{deg}(K)+1/2$. A key ingredient in the proof of this result is
to give asymptotics for the number of elliptic curves over an arbitrary number
field with a prescribed local condition; these results will follow from general
results for counting points of bounded height on weighted projective spaces
with a prescribed local condition.
