翻译 - 德语-英语 - friend

Even though its written in Present tense, this is often also used to indicate future intention - so "I'll write you an email" might be better.
I suppose 'I'm writing you an email' could theoretically be possible here as well, but it seems unlikely as its a bit of an obvious statement, if you're (hypothetically) already reading said email..!

Not 'I write to you an e-mail' but 'I write you an e-mail' or 'I'll write you an e-mail'

I'll write you an e-mail

I will write you an e-mail.
Warm regards

1) I'll write you an e-mail
2) Olympic Games