Here's a couple of arpeggio patterns:
E|------------3--7----------------------12--15--------------|
B|----------5------8-----------------12--------17-----------|
G|--------4----------9------------12--------------16--------|
D|------5--------------9-------14--------------------17-----|
A|----7------------------10--14-------------------------19--|
E|-----------------------------------------------------------|
This is jut a random thing to give you the general idea. Going up the first time is Em, going down is Em 1st inversion, going up again is Em 2nd inversion and finally going down again is Em. These are just chord tones played seperatly, nothing special. It really is easier to know the three notes of the Em triad and go from there. Memorizing all the patterns isn't really necessary.
About the modes... There are no Em modes. Modes are taken from the major, melodic minor and harmonic minor scales. Some are major, some are minor and some diminished.