Figure rising crime is court's raison d'être (6)
I believe the answer is:
tennis
'court's raison d'etre' is the definition.
I know that 'court' relates to this answer.
'figure rising crime' is the wordplay.
'figure' becomes 'ten' (ten is a number or figure).
'rising' says the letters should be written backwards (going up in a down clue).
'crime' becomes 'sin'.
'sin' back-to-front is 'nis'.
'ten'+'nis'='TENNIS'
'is' is the link.
Can you help me to learn more?
(Other definitions for tennis that I've seen before include "Table -, - elbow" , "Sent in to play the game, oddly" , "court contest" , "Sent in, oddly, to play a sport" , "Racquet sport" .)