Casual form:
Male speech: 僕の誕生日を覚えた! (Boku no tanjōbi o oboeta!)
Female speech: 私の誕生日を覚えた! (Watashi no tanjōbi o oboeta!)
Polite form:
Male speech: 僕の誕生日を覚えました! (Boku no tanjōbi o oboemashita!)
Female speech: 私の誕生日を覚えました! (Watashi no tanjōbi o oboemashita!)
All of these phrases say 'Remembered my birthday!', because it is really uncommon to say 'you are/you did', except when it is really unclear who you are referring to.
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The sentence 'do you remember me?' would be translated as ç§ã®ã“ã¨è¦šãˆã¦ã¾ã™ã‹in Japanese. This would be read as "watashi no koto oboetemasu ka".