Iwasawa theory
The main conjecture of Iwasawa theory was formulated as an assertion that two ways of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was eventually proved by Barry Mazur and Andrew Wiles for Q, and for all totally real number fields by Andrew Wiles. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).