Iwasawa theory
From this beginning, in the 1950s, a good-sized theory has been built up. A basic connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory might be able to move ahead finally from Kummer's century-old results on regular primes.