Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators. 


Is there a p-adic analogue of the story, that is a p-adic L-function $L\_p(C, s)$ and a conjecture about the special values? 


I have seen people study p-adic L-functions of modular forms, so I guess for elliptic curves the answer is positive, at least to the first question. What about more general curves? Is there a construction of $L\_p(C, s)?$



p-adic L-function of curves
To complete Chris's answer in comments, yes, we do expect such a $p$-adic $L$-function to exist, but we are far from being able to prove it. There are two problems: first we very likely need to show that the curve $C$, or equivalently its Jacobian, or equivalently its $L$-function $L(C,s)$ is automorphic, because all $p$-adic $L$-functions construction we have so far use in some fundamental way the deep properties of archimidean **automorphic** $L$-function. And proving the automorphy of a curve $C$ over $\mathbb Q$ (or for that matter, of an elliptic curve over a general number field) is wide open. Second, even when we know that $L(C,s)$ is $L(\pi,s)$ for nice automorphic representation $\pi$, current technologies does not allow yet to prove that the special values of $L(\pi,s)$ at integers are, conveniently normalized, algebraic numbers, let alone to interpolate those values $p$-adically into a $p$-adic $L$-function. There has been some progresses when $\pi$ is attached to a unitary group, and $p$ an ordinary prime for $\pi$ be the general case is still out of reach... 


