The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? .

I am aware of the statements of class field theory and the modularity theorem, as well as far-reaching generalizations that have to do with the conjectural Langlands group and motives. But on a basic level, I just don't understand why such statements should be true, other than that there is a lot of evidence that they are.

What is the philosophical impetus behind modularity results?

When I read about number theory, I can very easily understand the intuition behind ramification of primes (because the intuition is geometric), but as soon as we start talking about splitting of primes, and are therefore in the realm of modularity results, I lose all intuition of why things should be true (even though I can read and understand the results as an undergraduate can -- agreeing line by line).

An example of an answer for CFT can be the following thing I've heard, but was somewhat unsatisfied with because I didn't fully understand it: that it grew out of generalizations of Fourier analysis. (if you also think of this as the reason it's true, and can expatiate -- do!)

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? .

I am aware of the statements of class field theory and the modularity theorem, as well as far-reaching generalizations that have to do with the conjectural Langlands group and motives. But on a basic level, I just don't understand why such statements should be true, other than that there is a lot of evidence that they are.

What is the philosophical impetus behind modularity results?

When I read about number theory, I can very easily understand the intuition behind ramification of primes (because the intuition is geometric), but as soon as we start talking about splitting of primes, and are therefore in the realm of modularity results, I lose all intuition of why things should be true (even though I can read and understand the results as an undergraduate can -- agreeing line by line).

An example of an answer for CFT can be the following thing I've heard, but was somewhat unsatisfied with because I didn't fully understand it: that it grew out of generalizations of Fourier analysis. (if you also think of this as the reason it's true, and can expatiate -- do!)