In a small bowl, mix ½ cup of the sugar with the pectin.
Combine this mixture with the peaches, lemon juice and ascorbic acid in a medium saucepan.
Bring this mixture to a hard boil over high heat (meaning the mixture will not stop bubbling when stirred) as you continually stir.
Slowly add the remaining 3 ½ cups of sugar, a third at a time (so as not to reduce the temperature of the jam) until all sugar is dissolved before adding more.
Once you add all the sugar, bring the mixture back to a boil for one minute.
Remove from the heat, skim off any foam with a metal spoon, and carefully ladle into the sterilized jars, leaving ¼ inch of headspace.
Wipe the jar rims of any jam and screw on tops.
Prepare a hot water bath for canning. You can do this in a hot water bath canner or a deep pot with a cooling rack inside.
Immerse the jars (on top of the rack) in the boiling water to cover one to two inches about the top of the jar lids. Cover the pot with a lid and bring to a boil.
Process in the hot water bath for 10 minutes.
Carefully remove the jars from water; let stand at room temperature until cool.