This is true for the fields that the other farmers are leasing as well, but it won't be easy to rent these fields. Each field will be owned by a different farmer in town (ie Giorgio, Elise, or Fritz). In order to win over these fields, you'll have to partake in conquests. The conquests can be anything such as partaking in a yearly event, total earnings on a specific day, or varieties shipped on a specific day. Veronica will tell you when the days will be. If you lose a conquest, don't lose heart! You can always try again once a lease is up.