User:RobHar/Sandbox4

In number theory, p-adic L-functions arise in two different, but conjecturally related ways. The first way was pioneered by Kubota and Leopoldt in 1964 through their reinterpretation of the Kummer congruences as the existence of a continuous p-adic function interpolating the values of the Riemann zeta function. A p-adic L-function arising in this fashion is sometimes called an analytic p-adic L-function. A few years later?, Iwasawa constructed a p-adic L-function from the Selmer group?class group. These p-adic L-functions are sometimes referred to as arithmetic p-adic L-functions.

Elliptic curves and modular forms
In two papers, Yu Manin first constructed p-adic L-functions attached to modular forms. His results were restricted to level 1 and "slope" < 1. Yvette Amice and Jacques Velu generalised Manin's result to "slope" < k-1 (right?). Then, Misha M. Visik generalised this to include level Γ0(N) prime to p and a non-trivial nebentypus. Barry Mazur, John Tate, and Jeremy Teitelbaum improved the result to include p exactly dividing N.