This is the 10460th most frequent Russian word.
весомых
Significant; weighty.
The word 'весомыми' (short form of the plural instrumental case of 'весомый'), meaning 'weighty' or 'significant', is used here to describe the arguments.
Эти аргументы являются достаточно весомыми.
These arguments are quite weighty.
The adjective 'весомые', which translates to 'significant' or 'weighty', modifies 'доказательства' (evidence), showing the reliability or importance of the proof he provided.
Он предоставил весомые доказательства своей теории.
He provided significant evidence for his theory.
The term 'весомое' in 'весомое значение', meaning 'weighty or substantial meaning', emphasizes the impact the words have on the speaker.