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The famous Beilinson's conjecture predicts a relationship between the regulator map in $K$-theory and special value of $L$-function generalizing the Dirichlet's theorem in number theory. Please see this post for details.

In this paper by Tim Dokchitser, Rob de Jeu, Don Zagier, the authors construct families of hyperelliptic curves over $\mathbf{Q}$ of arbitrary genus g with (at least) g integral elements in $K_2$. With these elements they compute the regulator numerical and compare the regulator with special value of $L$-function of these curves which give numerical evidence of Beilinson's conjecture for $K_2$ of curves.

Here are my questions:

Is there explicitly formulated conjectures in p-adic fields and function fields as in number fields? Does one can do similar computations for curves over these fields? If not, what are the main obstacles?

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You may be interested in Colmez' Bourbaki survey "Fonctions L p-adiques", available here:

Section 2.8 is called "p-adic Beilinson conjectures".

See also his other survey (on the same page) called "La conjecture de Birch et Swinnerton-Dyer p-adique".

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