In Nekovar's introductory paper "Beilinson's Conjecture"


<http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf>


The conjecture is formulated for smooth projective varieties over $\mathbb{Q}$. However all the statements and proofs seem to equally work even for smooth projective varieties defined over an arbitrary number field $L$. Is this true? Are there any differences between the case over $\mathbb{Q}$ and number fields $L$?



Does Beilinson's conjecture on values L-functions work for smooth projective varieties over a number field
Yes, you can formulate Beilinson's conjectures for smooth projective varieties over a number field. I would recommend the following survey paper by Dinakar Ramakrishnan: *Regulators, algebraic cycles and values of $L$-functions*. Other references, with an emphasis on the equivariant version of the conjecture, include: Fontaine--Perrin-Riou, *Autour des conjectures de Bloch et Kato : cohomologie galoisienne et valeurs de fonctions $L$*, and the articles by Burns and Flach: *Motivic $L$-functions and Galois module structures; Tamagawa numbers for motives with (non-commutative) coefficients I, II*.


I should add that if $X$ is a smooth projective variety over a number field $K$ then its Weil restriction $Y=\operatorname{Res}\_{K/\mathbf{Q}} X$ exists and is a smooth projective variety over $\mathbf{Q}$, and the Beilinson conjectures for $X$ and $Y$ should be equivalent. I don't know a reference where this is worked out in detail, but see Fontaine, *Valeurs spéciales des fonctions $L$ des motifs*, Remarque 6.11 for some indications.


