Displaced pigeons must solve two problems: which way is which, and where am I relative to home. The compass problem is largely settled. Pigeons use the sun's azimuth corrected by an internal clock, demonstrable because shifting a bird's light cycle produces a predictable angular error in its departure bearing. Under overcast skies they fall back on magnetic cues.
The map problem is not settled. Italian experiments in which pigeons were made anosmic, or raised in air screened from natural odours, showed sharply impaired homing, supporting an olfactory gradient map. Other groups have argued that these manipulations disrupt attention and motivation as much as smell. Infrasound generated by terrain has also been implicated, notably in explaining release sites where pigeons reliably become disoriented.
Magnetoreception in pigeons has its own unresolved history. A widely cited candidate, iron-rich cells in the upper beak, was reinterpreted in 2012 when those cells were identified as macrophages rather than magnetosensors. Magnetic sensitivity in the birds is well supported behaviourally; the receptor has not been convincingly located.