To paraphrase John Wayne "The only good pigeon is a dead pigeon"!
it's legal to shoot wood pigeons as they are classified as a pest. The problem in our suburban area is that I suspect the neighbours would object. If its a rural allotment then I suspect it should be OK.
They need to be controlled as they are a breeding machine. They can breed all the year round if there is enough food.

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George the Pigman