The official Pokémon website has announced that players of Pokémon Brilliant Diamond and Pokémon Shining Pearl can get Arceus and Darkrai if they have a Pokémon Legends: Arceus save file on their Nintendo Switch system with all the main missions completed.

Get Arceus and Darkrai in Pokémon Brilliant Diamond and Pokémon Shining Pearl

Prove your Pokémon Legends: Arceus skills and obtain a special Mystery Gift item to encounter these Mythical Pokémon in Sinnoh.

A pair of Mythical Pokémon are waiting for you in the Sinnoh region! By following the instructions below, you can encounter Arceus and Darkrai in your Pokémon Brilliant Diamond and Pokémon Shining Pearl adventures. They're sure to make incredible additions to your team.

To find Arceus, you'll need to have a Pokémon Legends: Arceus save file on your Nintendo Switch system with all the main missions completed. You'll also need to enter the Hall of Fame and obtain the National Pokédex in Pokémon Brilliant Diamond or Pokémon Shining Pearl. Once all the conditions have been met, return to your room in Pokémon Brilliant Diamond or Pokémon Shining Pearl, where you can get the Azure Flute item. Then, go to the Spear Pillar at the top of Mount Coronet, and Arceus will reveal itself. Be ready to battle this powerful Pokémon!

To befriend Darkrai, you must first obtain the Member Card item—available between the new moons—via the Mystery Gift feature in Pokémon Brilliant Diamond or Pokémon Shining Pearl. Once again, you will also need to enter the Sinnoh region's Hall of Fame and obtain the National Pokédex. Do so, then travel to Newmoon Island, where Darkrai can be found.

Please note that to receive the Azure Flute in Pokémon Brilliant Diamond or Pokémon Shining Pearl, you will need to have downloaded the software update (version 1.3.0) scheduled for distribution on March 15, 2022, at 6:00 p.m. PDT.​

Source: https://www.pokemon.com/us/pokemon-news/get-arceus-and-darkrai-in-pokemon-brilliant-diamond-and-pokemon-shining-pearl/

Press Release - Mythical Pokémon Coming to Pokémon Brilliant Diamond and Pokémon Shining Pearl

The Pokémon Company shared today that a new update is coming to Pokémon Brilliant Diamond and Pokémon Shining Pearl. Players will be able to befriend Darkrai and, by playing Pokémon Legends: Arceus, fans will be able to encounter Arceus in Pokémon Brilliant Diamond and Pokémon Shining Pearl.

• Encountering Arceus: To encounter Arceus in Pokémon Brilliant Diamond and Pokémon Shining Pearl, players must update their game to version 1.3.0* and have a Pokémon Legends: Arceus save file on their Nintendo Switch system with all the main missions completed. They can then receive the Azure Flute item in Pokémon Brilliant Diamond or Pokémon Shining Pearl after entering the Hall of Fame and obtaining the National Pokédex. Once these conditions are met, players can go to the entrance of the Spear Pillar and the Mythical Pokémon Arceus will reveal itself.
• Befriending Darkrai:​ The Member Card item will be given as a gift from 8:24 a.m. CEST on Friday, 1 April, until 10:28 p.m. CEST on Saturday, 30 April 2022. After obtaining the item by selecting Get via Internet from the Mystery Gift feature in-game and fulfilling specific conditions—players must have entered the Hall of Fame, obtained the National Pokédex, and completed the in-game event for Cresselia — you will be able to go to Newmoon Island, where the Mythical Pokémon Darkrai can be found.​

The new update will be available starting Wednesday, March 16th, 2022.