This is the 15997th most frequent Russian word.
псевдониму
"Pseudonym" (a fictitious name used by an individual).
In this sentence, 'псевдонимом' indicates the tool or means by which the author signed the document, which is the pseudonym.
Автор подписал документ псевдонимом.
The author signed the document with a pseudonym.
Here, 'псевдониму' conveys the concept of using the pseudonym as a basis for determining something, demonstrating its function as a characteristic.
По псевдониму можно определить жанр произведения.
By the pseudonym, one can determine the genre of the work.
In this context, 'псевдоним' refers to the actual pseudonym as an object or concept that provides anonymity.