  Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$
which appears in the work of Nagao, Rosen, and Silverman, we conjecture that
100% of elliptic surfaces have rank $0$ when ordered by the size of the
coefficients of their Weierstrass equations, and present a probabilistic
heuristic to justify this conjecture. We then discuss how it would follow from
either understanding of certain $L$-functions, or from understanding of the
local behaviour of the surfaces. Finally, we make a conjecture about ranks of
elliptic surfaces over finite fields, and highlight some experimental evidence
supporting it.
