This is my own build of Ezreal as an ap mage. This build focuses on late game dominance and kiting, with extremely short cooldowns. I'm not too good at guides, so this will be short and sweet.

In the early game, you will want to use your Mystic Shot to farm and harass the opponent in lane. Focus on farming more than poking, and make sure to stack the Tear of the Goddess and rush Archangel's Staff asap, as this will help you spam and get to your damage quicker.

After you get your Trueshot Barrage, you should be looking to snipe low health enemies or engage on your mid lane opponent, as your damage will spike if you land it. Along with Mystic Shot's passive, his ult is on a very short cooldown, especially if you get Iceborn Gauntlet early on. Rushing Lich Bane will make your Mystic Shot hurt quite a bit, as it procs the lich bane passive. Using Trueshot Barrage to farm other lanes or even your own while they are away will help you get items sooner. Also, it is great for stealing dragon and baron.

Once you get your items, you will be kiting like crazy. Your mana will be off the charts, and you will have at least 30% cooldown reduction, making your Mystic Shot have almost no cooldown, while dealing a ton of damage. Note that Archangel's Staff has a ridiculous amount of ability power when you have all your items. Your ultimate will hit like a truck in teamfights, as will your w if you can land it on multiple targets. Once you have Iceborn Gauntlet you should hardly be dying unless you get cc'd heavily. Additionally, try to peel of any charging enemies coming at you or your Marksmen, as you will be in roughly the same position in teamfights.

Glacial Ezreal was made entirely by me, and I believe he can be very viable in the mid lane. His damage potential is very high and his kiting potential is ridiculous. If you try out this build, tell me what you think and if you like it. Just remember that you are pretty squishy, but your Arcane Shift will get you out of quite a bit of danger. Try to keep Arcane Shift up as much as possible, and spam Mystic Shot to keep your Trueshot Barrage and other abilities on low cooldown.