My question is how should one think of p-adic L functions? I know they have been constructed classically by interpolating values of complex L-functions. Recently I have seen people think about them in terms of Euler systems. But we know only a few Euler systems and there are lot of p-adic L functions. In case of elliptic curves(at least over $\mathbbQ$) complex L-functions give information about the Galois representations. Should the p-adic L-function give some information about some p-adic Galois representation? It seems to be the case in case of cyclotomic fields where we think of the cyclotomic character as a 1-dimensional representation. I apologize in advance if my questions are vague. I am just starting to learn about the subject.

My question is how should one think of p-adic L functions? I know they have been constructed classically by interpolating values of complex L-functions. Recently I have seen people think about them in terms of Euler systems. But we know only a few Euler systems and there are lot of p-adic L functions. In case of elliptic curves(at least over $\mathbb{Q}$) complex L-functions give information about the Galois representations. Should the p-adic L-function give some information about some p-adic Galois representation? It seems to be the case in case of cyclotomic fields where we think of the cyclotomic character as a 1-dimensional representation. I apologize in advance if my questions are vague. I am just starting to learn about the subject.