Generally speaking, no. While theoretically a JPEG could be used to STORE a virus, the chances of that code actually running on your computer is almost nil. (Provided you don't do silly things such as clicking on a file named "img.jpeg.exe" or what have you.) Of course from time to time in the past there have been bugs. Some of the early Windows versions for example. AFAIK most (if not all) JPEG decoders in common use today have already ironed out those issues.

JPEG as a format allows you to add extra data to the end. ZIP as a format allows you to add extra data to the beginning. So its probably easy to make a JAR-JPEG polygot.

Generally though, even if you make an executable JPEG, the thing stopping this is that normally you look at the extension to decide how to interpret a file, so when you see a .jpg ending, you don't try and execute.

You could store the bytes of an EXE inside an image, just like how I could paste the base-64 of an EXE on the end of this message. The trick is going from that to actually running it.

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Only if there is a flaw in the jpeg decoder