We discuss in this paper the influence of the presence of an elevated strike object on the peak of the lightning return stroke current determined from remote field measurements. We develop analytical expressions relating the lightning return stroke channel-base current and the far electromagnetic field for different specific cases, namely, (1) ground-initiated return strokes (classical transmission line (TL) model), (2) ground-initiated return strokes including possible reflections at ground level, (3) tall strike objects for which the current's zero-to-peak time is smaller than the travel time along the object, and (4) electrically short strike objects. It is shown that for tall structures, the field enhancement relative to a return stroke initiated at ground level is expressed through a factor equal to k_tall = [1 + c / v (1 - 2 ρ_t)] / (1 - ρ_t), where v and c are the return stroke front speed and the speed of light in vacuum, respectively, and ρ_t is the top reflection coefficient. For very short towers and/or very slow return stroke current wavefronts, when the condition t_f very much greater than h / c applies, expressions relating the far electromagnetic field and the return stroke current were also derived. For case (2), return strokes initiated at ground level, the field enhancement relative to a return stroke initiated at ground level, case (1), is expressed through a factor equal to k_short = (1 + (c / v) ρ_{ch - g}) / (1 + ρ_{ch - g}), where ρ_ch-g represents the reflection coefficient between the lightning channel and the grounding impedance.