Hello kasakie!

Everything you need to solve this little mystery is in Rabanastre.

Talk to Filo down in Lowtown. Then talk to the woman at the fountain. Head to the Muthru Bazaar and find a Bangaa who has some information about the necklace. From there you'll head to Yugri's Magicks to find an Imperial soldier and then back to Lowtown to talk to Kytes. Find Filo again and then end up back at Yugri's.