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If Ashiok's Erasure has exiled an adventurer card, your opponents can't cast spells with the same name as the creature, but may cast their Adventures.
The exiled spell isn't countered, but it won't resolve. This works against spells that can't be countered.
If Ashiok's Erasure has exiled a split card, your opponents can't cast a spell with the same name as either half of that card.
If Ashiok's Erasure leaves the battlefield before its enters-the-battlefield ability has resolved, its last ability triggers and resolves with no effect, then its enters-the-battlefield ability exiles the target spell for the rest of the game.
If there is no exiled card (perhaps because the exiled spell was a copy of a spell), or if the exiled card leaves exile somehow, Ashiok's Erasure won't stop players from casting spells.
