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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 $L_p(C, s)?$

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  • $\begingroup$ Partial information related to the question can be found in the MO-questions mathoverflow.net/questions/13287 and mathoverflow.net/questions/168427 $\endgroup$
    – Matthias Wendt
    Sep 19, 2014 at 8:38
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    $\begingroup$ The complex $L(C,s)$ can be written in terms of the Jacobian of $C$. So you might also ask if for all abelian varieties $A/\mathbb{Q}$ there is a $p$-adic $L$-function. Certainly there should be one, but we do not know how to construct it in general, I fear. For instance, we may not know anything about the "modularity" of $A$ and hence not even that $L(C,s)$ extends to the complex plane and in particular to the places where we would like to interpolate $p$-adically. $\endgroup$
    – Chris Wuthrich
    Sep 19, 2014 at 9:27

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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...

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