Preheat the oven to 400 degrees F and line a rimmed baking sheet with parchment.
Cut the peppers lengthwise and remove the stems, seeds and ribs. If needed, trim a small amount off the bottom of each pepper half so that it will lie flat when cut-side up (making them steady is important so the filling stays intact during baking). Chop the trimmings into small pieces and set aside.
Toss the pepper halves in a medium bowl with the olive oil, 1/2 teaspoon salt and several grinds of black pepper. Set aside.
Stir together the softened cream cheese and reserved pepper trimmings in another medium bowl until smooth. Gently fold in the crabmeat and season with salt and pepper.
Fill each pepper half with about 1 tablespoon of the crab mixture so it comes over the top of the pepper slightly. Place on the prepared baking sheet and bake until the filling is beginning to brown, about 18 minutes. Serve warm.
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