Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to deal with than those over number fields. Can someone actually elaborate on this analogy between number fields and function fields? I'm not sure where I can find information about this. Ring of integers being Dedekind domains, finite residue field, RH over function fields easier to deal with, anything else? Being quite ignorant about this analogy, I am actually not even convinced that why working over function fields "should" give insights about questions about number fields.